{"id":101696,"date":"2026-09-07T09:01:13","date_gmt":"2026-09-07T01:01:13","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/101696.html"},"modified":"2026-09-07T09:01:13","modified_gmt":"2026-09-07T01:01:13","slug":"%e3%80%90%e7%ae%97%e6%b3%95%e8%ae%be%e8%ae%a1%e4%b8%8e%e5%88%86%e6%9e%90%e3%80%91%e5%bc%80%e5%ad%a6%e8%af%be%e5%89%8d%e9%a2%84%e4%b9%a0%e5%85%a8%e6%94%bb%e7%95%a5%ef%bc%9a%e4%bb%8e%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/101696.html","title":{"rendered":"\u3010\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5206\u6790\u3011\u5f00\u5b66\u8bfe\u524d\u9884\u4e60\u5168\u653b\u7565\uff1a\u4ece\u7b97\u6cd5\u57fa\u672c\u6982\u5ff5\u5230\u65f6\u95f4\/\u7a7a\u95f4\u590d\u6742\u5ea6\u5206\u6790\uff08\u4e07\u5b57\u8be6\u89e3\uff0c\u542b\u4f8b\u9898\u7cbe\u8bb2\u4e0e\u8003\u70b9\u603b\u7ed3\uff09"},"content":{"rendered":"<h2>\u3010\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5206\u6790\u3011\u5f00\u5b66\u8bfe\u524d\u9884\u4e60\u5168\u653b\u7565&#xff1a;\u4ece\u7b97\u6cd5\u57fa\u672c\u6982\u5ff5\u5230\u65f6\u95f4\/\u7a7a\u95f4\u590d\u6742\u5ea6\u5206\u6790&#xff08;\u4e07\u5b57\u8be6\u89e3&#xff0c;\u542b\u4f8b\u9898\u7cbe\u8bb2\u4e0e\u8003\u70b9\u603b\u7ed3&#xff09;<\/h2>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260907010111-6a9e0cd74d885.jpg\" alt=\"\u5728\u8fd9\u91cc\u63d2\u5165\u56fe\u7247\u63cf\u8ff0\" \/><\/p>\n<p>&#x1f4cc; \u5199\u5728\u524d\u9762&#xff1a;\u65b0\u5b66\u671f\u300a\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5206\u6790\u300b\u8bfe\u7a0b\u5373\u5c06\u5f00\u59cb&#xff0c;\u5f88\u591a\u540c\u5b66\u4e00\u542c\u5230&#034;\u7b97\u6cd5\u5206\u6790&#034;\u56db\u4e2a\u5b57\u5c31\u5934\u76ae\u53d1\u9ebb\u3002\u5176\u5b9e&#xff0c;\u8fd9\u95e8\u8bfe\u7684\u7b2c\u4e00\u7ae0\u5185\u5bb9\u2014\u2014\u7b97\u6cd5\u7684\u57fa\u672c\u6982\u5ff5\u3001\u7b97\u6cd5\u7684\u4e94\u5927\u7279\u5f81\u3001\u6e10\u8fdb\u7b26\u53f7&#xff08;\u5927O\u3001\u5927\u03a9\u3001\u5927\u0398&#xff09;\u3001\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u7a7a\u95f4\u590d\u6742\u5ea6\u5206\u6790\u2014\u2014\u662f\u6574\u4e2a\u8bfe\u7a0b\u7684\u5730\u57fa\u3002\u5730\u57fa\u6253\u7262\u4e86&#xff0c;\u540e\u9762\u5b66\u5206\u6cbb\u3001\u8d2a\u5fc3\u3001\u52a8\u6001\u89c4\u5212\u3001\u56de\u6eaf\u3001\u5206\u652f\u9650\u754c\u90fd\u4f1a\u4e8b\u534a\u529f\u500d\u3002<\/p>\n<p>\u672c\u6587\u57fa\u4e8e\u6559\u6750\u7b2c\u4e00\u7ae0\u7684\u77e5\u8bc6\u70b9&#xff0c;\u6309\u7167&#034;\u6982\u5ff5\u56de\u987e \u2192 \u4f8b\u9898\u7cbe\u8bb2 \u2192 \u9519\u8bef\u5256\u6790 \u2192 \u65b9\u6cd5\u603b\u7ed3 \u2192 \u8003\u70b9\u63d0\u70bc&#034;\u7684\u8109\u7edc&#xff0c;\u628a\u8bfe\u524d\u9884\u4e60\u5185\u5bb9\u5b8c\u6574\u5730\u68b3\u7406\u4e86\u4e00\u904d\u3002\u5168\u6587\u914d\u6709\u4ee3\u7801\u3001\u516c\u5f0f\u63a8\u5bfc\u3001\u5bf9\u6bd4\u8868\u683c\u548c\u6613\u9519\u70b9\u63d0\u793a&#xff0c;\u9002\u5408\u9884\u4e60\u3001\u590d\u4e60\u548c\u8003\u524d\u7a81\u51fb\u4f7f\u7528\u3002\u5efa\u8bae\u6536\u85cf\u540e\u6162\u6162\u6d88\u5316\u3002&#x1f447;<\/p>\n<hr \/>\n<h3>\u76ee\u5f55<\/h3>\n<li>\u6982\u8ff0&#xff1a;\u4e3a\u4ec0\u4e48\u8981\u5b66\u7b97\u6cd5<\/li>\n<li>\u7b97\u6cd5\u7684\u6982\u5ff5\u4e0e\u6b63\u786e\u6027<\/li>\n<li>\u7b97\u6cd5\u8bbe\u8ba1\u5e94\u8ffd\u6c42\u7684\u516d\u5927\u76ee\u6807<\/li>\n<li>\u3010\u4f8b1-1\u3011\u5355\u94fe\u8868\u67e5\u627e\u7b97\u6cd5\u7684\u9519\u8bef\u5206\u6790&#xff08;\u7ecf\u5178\u4f8b\u9898&#xff09;<\/li>\n<li>\u7b97\u6cd5\u7684\u4e94\u5927\u7279\u5f81&#xff08;Knuth\u5b9a\u4e49&#xff09;<\/li>\n<li>\u6848\u4f8b&#xff1a;\u6c42 6x&#043;5y&#043;4z&#061;50 \u7684\u6b63\u6574\u6570\u89e3&#xff08;\u81ea\u7136\u8bed\u8a00\u63cf\u8ff0\u7b97\u6cd5&#xff09;<\/li>\n<li>\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5&#xff08;\u8f97\u8f6c\u76f8\u9664\u6cd5&#xff09;\u4e0e\u7279\u5f81\u8fa8\u6790\u586b\u7a7a\u9898<\/li>\n<li>\u3010\u4f8b1-2\u3011\u8fdd\u53cd\u7b97\u6cd5\u7279\u5f81\u7684\u4e24\u6bb5\u4ee3\u7801<\/li>\n<li>\u63cf\u8ff0\u7b97\u6cd5\u7684\u4e09\u79cd\u5e38\u7528\u65b9\u6cd5<\/li>\n<li>\u7528C\/C&#043;&#043;\u63cf\u8ff0\u7b97\u6cd5\u7684\u4e00\u822c\u5f62\u5f0f<\/li>\n<li>\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u8054\u7cfb\u4e0e\u533a\u522b<\/li>\n<li>1.2 \u7b97\u6cd5\u7684\u5206\u6790&#xff1a;\u65f6\u95f4\u590d\u6742\u5ea6\u6982\u8ff0<\/li>\n<li>\u6e10\u8fdb\u7b26\u53f7&#xff1a;\u5927O\u3001\u5927\u03a9\u3001\u5927\u0398 \u8be6\u89e3<\/li>\n<li>\u3010\u4f8b1-3\u3011\u4e09\u91cd\u5faa\u73af\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790&#xff08;\u542b\u5b8c\u6574\u6c42\u548c\u63a8\u5bfc&#xff09;<\/li>\n<li>\u7b97\u6cd5\u7684\u6700\u597d\u3001\u6700\u574f\u548c\u5e73\u5747\u60c5\u51b5<\/li>\n<li>\u3010\u4f8b1-4\u3011\u987a\u5e8f\u67e5\u627e\u7684\u4e09\u79cd\u60c5\u51b5\u5206\u6790&#xff08;\u91cd\u70b9&#043;\u96be\u70b9&#xff09;<\/li>\n<li>\u975e\u9012\u5f52\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790 &#043; \u4f8b1-5<\/li>\n<li>\u9012\u5f52\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790 &#043; \u4f8b1-6&#xff08;\u5f52\u5e76\u6392\u5e8f&#xff09;<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6\u5206\u6790 &#043; \u4f8b1-7\u3001\u4f8b1-8<\/li>\n<li>\u5168\u7ae0\u6613\u9519\u70b9\u5927\u76d8\u70b9<\/li>\n<li>\u8003\u70b9\u603b\u7ed3\u4e0e\u5b66\u4e60\u5efa\u8bae<\/li>\n<hr \/>\n<h3>\u4e00\u3001\u6982\u8ff0&#xff1a;\u4e3a\u4ec0\u4e48\u8981\u5b66\u7b97\u6cd5<\/h3>\n<p>\u5728\u6b63\u5f0f\u8fdb\u5165\u65b0\u77e5\u8bc6\u4e4b\u524d&#xff0c;\u6211\u4eec\u5148\u56de\u5fc6\u4e00\u4e0b&#xff1a;\u7b97\u6cd5\u7684\u5b9a\u4e49\u3001\u7b97\u6cd5\u7684\u57fa\u672c\u6b65\u9aa4&#xff0c;\u8fd9\u4e9b\u5185\u5bb9\u5176\u5b9e\u6211\u4eec\u5728\u300a\u6570\u636e\u7ed3\u6784\u300b\u8bfe\u7a0b\u4e2d\u90fd\u5b66\u8fc7\u3002\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5206\u6790\u8fd9\u95e8\u8bfe\u5e76\u4e0d\u662f\u4ece\u96f6\u5f00\u59cb&#xff0c;\u800c\u662f\u7ad9\u5728\u6570\u636e\u7ed3\u6784\u7684\u80a9\u8180\u4e0a&#xff0c;\u628a&#034;\u5982\u4f55\u8bbe\u8ba1\u4e00\u4e2a\u597d\u7b97\u6cd5\u3001\u5982\u4f55\u8bc4\u4ef7\u4e00\u4e2a\u7b97\u6cd5\u7684\u597d\u574f&#034;\u8fd9\u4e24\u4ef6\u4e8b\u8bb2\u5f97\u66f4\u7cfb\u7edf\u3001\u66f4\u6570\u5b66\u5316\u3002<\/p>\n<p>\u4e3a\u4ec0\u4e48\u503c\u5f97\u4e13\u95e8\u5f00\u4e00\u95e8\u8bfe\u6765\u7814\u7a76\u7b97\u6cd5&#xff1f;\u56e0\u4e3a\u5728\u5b9e\u9645\u8f6f\u4ef6\u5f00\u53d1\u4e2d&#xff0c;\u6211\u4eec\u7ecf\u5e38\u9047\u5230\u8fd9\u6837\u7684\u573a\u666f&#xff1a;<\/p>\n<ul>\n<li>\u540c\u4e00\u4e2a\u95ee\u9898&#xff0c;\u7532\u5199\u51fa\u6765\u7684\u7a0b\u5e8f\u8dd11\u79d2&#xff0c;\u4e59\u5199\u51fa\u6765\u7684\u7a0b\u5e8f\u8dd11\u5c0f\u65f6\u2014\u2014\u5dee\u8ddd\u5728\u54ea\u91cc&#xff1f;\u7b97\u6cd5\u3002<\/li>\n<li>\u6570\u636e\u91cf\u4ece1\u4e07\u6761\u6da8\u5230100\u4e07\u6761&#xff0c;\u539f\u6765\u7684\u7a0b\u5e8f\u7a81\u7136&#034;\u8dd1\u4e0d\u52a8&#034;\u4e86\u2014\u2014\u4e3a\u4ec0\u4e48&#xff1f;\u590d\u6742\u5ea6\u589e\u957f\u7684\u9636\u4e0d\u540c\u3002<\/li>\n<li>\u9762\u8bd5\u65f6\u624b\u5199\u4ee3\u7801&#xff0c;\u9762\u8bd5\u5b98\u95ee&#034;\u4f60\u8fd9\u4e2a\u89e3\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u591a\u5c11&#xff1f;\u80fd\u4e0d\u80fd\u4f18\u5316&#xff1f;&#034;\u2014\u2014\u8003\u7684\u8fd8\u662f\u7b97\u6cd5\u5206\u6790\u80fd\u529b\u3002<\/li>\n<\/ul>\n<p>\u6240\u4ee5&#xff0c;\u5b66\u4e60\u8fd9\u95e8\u8bfe\u7684\u7b2c\u4e00\u4e2a\u5fc3\u6001\u8f6c\u53d8\u662f&#xff1a;\u4e0d\u8981\u53ea\u6ee1\u8db3\u4e8e&#034;\u80fd\u5199\u51fa\u6765\u3001\u80fd\u8dd1\u901a&#034;&#xff0c;\u800c\u8981\u8ffd\u6c42&#034;\u5199\u5f97\u597d\u3001\u8dd1\u5f97\u5feb\u3001\u8bf4\u5f97\u6e05&#034;\u3002&#034;\u8bf4\u5f97\u6e05&#034;\u5c31\u662f\u6307\u4f60\u80fd\u7528\u6e10\u8fdb\u7b26\u53f7\u51c6\u786e\u5730\u5206\u6790\u51fa\u81ea\u5df1\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6&#xff0c;\u8fd9\u662f\u533a\u5206&#034;\u7801\u519c&#034;\u548c&#034;\u5de5\u7a0b\u5e08&#034;\u7684\u91cd\u8981\u5206\u6c34\u5cad\u3002<\/p>\n<hr \/>\n<h3>\u4e8c\u3001\u7b97\u6cd5\u7684\u6982\u5ff5\u4e0e\u6b63\u786e\u6027<\/h3>\n<h4>2.1 \u7b97\u6cd5\u7684\u5b9a\u4e49<\/h4>\n<p>\u7b97\u6cd5&#xff08;Algorithm&#xff09;&#xff1a;\u7b97\u6cd5\u662f\u6c42\u89e3\u95ee\u9898\u7684\u4e00\u7cfb\u5217\u7b97\u6cd5\u6b65\u9aa4&#xff0c;\u7528\u6765\u5c06\u8f93\u5165\u6570\u636e\u8f6c\u6362\u6210\u8f93\u51fa\u7ed3\u679c\u3002<\/p>\n<p>\u8fd9\u4e2a\u5b9a\u4e49\u4e2d\u6709\u4e09\u4e2a\u5173\u952e\u8bcd&#xff1a;<\/p>\n<li>\u4e00\u7cfb\u5217\u6b65\u9aa4\u2014\u2014\u7b97\u6cd5\u662f\u6709\u5e8f\u7684\u3001\u53ef\u6267\u884c\u7684\u6307\u4ee4\u5e8f\u5217&#xff0c;\u4e0d\u662f\u4e00\u53e5\u7a7a\u6cdb\u7684\u601d\u60f3&#xff1b;<\/li>\n<li>\u8f93\u5165\u6570\u636e\u2014\u2014\u7b97\u6cd5\u5904\u7406\u7684\u539f\u6750\u6599&#xff1b;<\/li>\n<li>\u8f93\u51fa\u7ed3\u679c\u2014\u2014\u7b97\u6cd5\u8981\u4ea4\u4ed8\u7684\u6210\u54c1\u3002<\/li>\n<p>\u4e00\u53e5\u8bdd\u6982\u62ec&#xff1a;\u7b97\u6cd5 &#061; \u4ece\u8f93\u5165\u5230\u8f93\u51fa\u7684\u786e\u5b9a\u6027\u53d8\u6362\u8fc7\u7a0b\u3002<\/p>\n<h4>2.2 \u7b97\u6cd5\u7684\u6b63\u786e\u6027<\/h4>\n<p>\u5982\u679c\u4e00\u4e2a\u7b97\u6cd5\u5bf9\u5176\u6bcf\u4e00\u4e2a\u8f93\u5165\u5b9e\u4f8b&#xff0c;\u90fd\u80fd\u8f93\u51fa\u6b63\u786e\u7684\u7ed3\u679c\u5e76\u505c\u6b62&#xff0c;\u5219\u79f0\u5b83\u662f\u6b63\u786e\u7684\u3002<\/p>\n<p>\u6ce8\u610f\u8fd9\u91cc\u6709\u4e24\u4e2a\u6761\u4ef6&#xff0c;\u7f3a\u4e00\u4e0d\u53ef&#xff1a;<\/p>\n<table>\n<tr>\u6761\u4ef6\u542b\u4e49\u53cd\u4f8b<\/tr>\n<tbody>\n<tr>\n<td>\u7ed3\u679c\u6b63\u786e<\/td>\n<td>\u5bf9\u4efb\u4f55\u5408\u6cd5\u8f93\u5165&#xff0c;\u8f93\u51fa\u90fd\u7b26\u5408\u95ee\u9898\u8981\u6c42<\/td>\n<td>\u67e5\u627e\u7b97\u6cd5\u628a\u7b2c3\u4e2a\u5143\u7d20\u9519\u62a5\u6210\u7b2c2\u4e2a<\/td>\n<\/tr>\n<tr>\n<td>\u80fd\u591f\u505c\u6b62<\/td>\n<td>\u7b97\u6cd5\u5fc5\u987b\u5728\u6709\u9650\u6b65\u5185\u7ed3\u675f&#xff0c;\u4e0d\u80fd\u6b7b\u5faa\u73af<\/td>\n<td>while(1) \u91cc\u6c38\u8fdc\u6ee1\u8db3\u7684\u6761\u4ef6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u26a0\ufe0f \u7279\u522b\u6ce8\u610f&#xff1a;&#034;\u6bcf\u4e00\u4e2a\u8f93\u5165\u5b9e\u4f8b&#034;\u662f\u5168\u79f0\u91cf\u8bcd\u3002\u4e00\u4e2a\u7b97\u6cd5\u53ea\u8981\u5b58\u5728\u54ea\u6015\u4e00\u4e2a\u8f93\u5165\u4f7f\u5b83\u8f93\u51fa\u9519\u8bef\u7ed3\u679c\u6216\u8005\u65e0\u6cd5\u7ec8\u6b62&#xff0c;\u5b83\u5c31\u4e0d\u662f\u6b63\u786e\u7684\u7b97\u6cd5\u3002\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u6211\u4eec\u6d4b\u8bd5\u7a0b\u5e8f\u65f6\u8981\u8bbe\u8ba1\u8fb9\u754c\u7528\u4f8b&#xff08;\u7a7a\u8f93\u5165\u3001\u6781\u7aef\u503c\u3001\u975e\u6cd5\u503c&#xff09;\u2014\u2014\u56e0\u4e3a\u9b54\u9b3c\u5f80\u5f80\u85cf\u5728\u8fb9\u754c\u91cc\u3002<\/p>\n<h4>2.3 \u7b97\u6cd5\u8d28\u91cf\u7684\u8bc4\u4ef7\u7ef4\u5ea6<\/h4>\n<p>\u540c\u4e00\u95ee\u9898\u53ef\u7528\u4e0d\u540c\u7b97\u6cd5\u89e3\u51b3&#xff0c;\u800c\u4e00\u4e2a\u7b97\u6cd5\u7684\u8d28\u91cf\u4f18\u52a3\u5c06\u5f71\u54cd\u5230\u7b97\u6cd5\u4e43\u81f3\u7a0b\u5e8f\u7684\u6548\u7387\u3002\u7b97\u6cd5\u5206\u6790\u7684\u76ee\u7684\u5728\u4e8e&#xff1a;\u9009\u62e9\u5408\u9002\u7684\u7b97\u6cd5 &#043; \u6539\u8fdb\u5df2\u6709\u7b97\u6cd5\u3002<\/p>\n<p>\u5bf9\u4e00\u4e2a\u7b97\u6cd5\u7684\u8bc4\u4ef7&#xff0c;\u4e3b\u8981\u4ece\u4e24\u4e2a\u7ef4\u5ea6\u8003\u8651&#xff1a;<\/p>\n<ul>\n<li>\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;\u7b97\u6cd5\u6267\u884c\u6240\u9700\u65f6\u95f4\u968f\u95ee\u9898\u89c4\u6a21\u589e\u957f\u7684\u8d8b\u52bf&#xff1b;<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;\u7b97\u6cd5\u6267\u884c\u6240\u9700\u5b58\u50a8\u7a7a\u95f4\u968f\u95ee\u9898\u89c4\u6a21\u589e\u957f\u7684\u8d8b\u52bf\u3002<\/li>\n<\/ul>\n<p>\u8fd9\u4e24\u4e2a\u7ef4\u5ea6\u5c06\u5728\u672c\u6587\u7b2c12\u8282\u4e4b\u540e\u8be6\u7ec6\u5c55\u5f00&#xff0c;\u5b83\u4eec\u662f\u672c\u8bfe\u7a0b\u7b2c\u4e00\u7ae0\u7684\u6838\u5fc3&#xff0c;\u4e5f\u662f\u671f\u672b\u8003\u8bd5\u7684\u91cd\u4e2d\u4e4b\u91cd\u3002<\/p>\n<hr \/>\n<h3>\u4e09\u3001\u7b97\u6cd5\u8bbe\u8ba1\u5e94\u8ffd\u6c42\u7684\u516d\u5927\u76ee\u6807<\/h3>\n<p>\u6559\u6750\u4e2d\u6307\u51fa&#xff0c;\u7b97\u6cd5\u8bbe\u8ba1\u5e94\u5f53\u8ffd\u6c42\u6ee1\u8db3\u4ee5\u4e0b\u76ee\u6807&#xff0c;\u800c\u4e0d\u662f\u53ea\u77e5\u9053\u6982\u5ff5\u2014\u2014\u53ea\u77e5\u9053\u6982\u5ff5\u800c\u4e0d\u52a8\u624b\u5b9e\u8df5&#xff0c;\u6c38\u8fdc\u4e0d\u4f1a\u8fdb\u6b65&#xff1a;<\/p>\n<p>\u6b63\u786e\u6027\u3001\u53ef\u4f7f\u7528\u6027\u3001\u53ef\u8bfb\u6027\u3001\u5065\u58ee\u6027\u3001\u9ad8\u6548\u7387\u3001\u4f4e\u5b58\u50a8\u91cf\u9700\u6c42<\/p>\n<p>\u6211\u4eec\u9010\u6761\u89e3\u91ca&#xff1a;<\/p>\n<h4>3.1 \u6b63\u786e\u6027&#xff08;Correctness&#xff09;<\/h4>\n<p>\u7b97\u6cd5\u5fc5\u987b\u80fd\u591f\u6b63\u786e\u89e3\u51b3\u95ee\u9898&#xff0c;\u5bf9\u5408\u6cd5\u7684\u8f93\u5165\u4ea7\u751f\u7b26\u5408\u8981\u6c42\u7684\u8f93\u51fa\u3002\u8fd9\u662f\u6700\u57fa\u672c\u7684\u8981\u6c42&#xff0c;\u4e00\u4e2a\u7ed3\u679c\u9519\u8bef\u7684\u7b97\u6cd5&#xff0c;\u8dd1\u5f97\u518d\u5feb\u4e5f\u662f\u5e9f\u7269\u3002<\/p>\n<h4>3.2 \u53ef\u4f7f\u7528\u6027&#xff08;Usability&#xff09;<\/h4>\n<p>\u7b97\u6cd5\u5e94\u5f53\u4fbf\u4e8e\u4f7f\u7528&#xff0c;\u63a5\u53e3\u6e05\u6670&#xff0c;\u4f7f\u7528\u8005\u4e0d\u9700\u8981\u4e86\u89e3\u5185\u90e8\u5b9e\u73b0\u7ec6\u8282\u5c31\u80fd\u6b63\u786e\u8c03\u7528\u3002\u6bd4\u5982\u51fd\u6570\u7684\u53c2\u6570\u542b\u4e49\u660e\u786e\u3001\u8fd4\u56de\u503c\u7ea6\u5b9a\u6e05\u695a\u3001\u6709\u4f7f\u7528\u6587\u6863\u6216\u6ce8\u91ca\u3002<\/p>\n<h4>3.3 \u53ef\u8bfb\u6027&#xff08;Readability&#xff09;<\/h4>\n<p>\u4ee3\u7801\u9996\u5148\u662f\u5199\u7ed9\u4eba\u770b\u7684&#xff0c;\u5176\u6b21\u624d\u662f\u7ed9\u673a\u5668\u6267\u884c\u7684\u3002\u547d\u540d\u89c4\u8303\u3001\u7ed3\u6784\u6e05\u6670\u3001\u9002\u5f53\u6ce8\u91ca\u7684\u7b97\u6cd5&#xff0c;\u4fbf\u4e8e\u4ed6\u4eba\u7406\u89e3\u3001\u7ef4\u62a4\u548c\u4fee\u6539\u3002\u4e00\u4e2a\u8c01\u4e5f\u770b\u4e0d\u61c2\u7684&#034;\u70ab\u6280&#034;\u4ee3\u7801&#xff0c;\u5728\u56e2\u961f\u534f\u4f5c\u4e2d\u662f\u8d1f\u8d44\u4ea7\u3002<\/p>\n<h4>3.4 \u5065\u58ee\u6027&#xff08;Robustness&#xff09;<\/h4>\n<p>\u5f53\u8f93\u5165\u975e\u6cd5\u6570\u636e\u6216\u51fa\u73b0\u610f\u5916\u60c5\u51b5&#xff08;\u5982\u7a7a\u6307\u9488\u3001\u9664\u96f6\u3001\u8d8a\u754c&#xff09;\u65f6&#xff0c;\u7b97\u6cd5\u80fd\u505a\u51fa\u6070\u5f53\u7684\u53cd\u5e94\u6216\u8fdb\u884c\u5904\u7406&#xff0c;\u800c\u4e0d\u662f\u83ab\u540d\u5176\u5999\u5730\u5d29\u6e83\u3002\u5065\u58ee\u6027\u8981\u6c42\u6211\u4eec\u5199\u4ee3\u7801\u65f6\u65f6\u523b\u601d\u8003&#xff1a;\u201c\u5982\u679c\u8fd9\u91cc\u51fa\u9519\u4e86\u600e\u4e48\u529e&#xff1f;\u201d<\/p>\n<h4>3.5 \u9ad8\u6548\u7387&#xff08;High Efficiency&#xff09;<\/h4>\n<p>\u6307\u7b97\u6cd5\u6267\u884c\u901f\u5ea6\u5feb&#xff0c;\u5373\u65f6\u95f4\u590d\u6742\u5ea6\u4f4e\u3002<\/p>\n<h4>3.6 \u4f4e\u5b58\u50a8\u91cf\u9700\u6c42&#xff08;Low Storage Requirement&#xff09;<\/h4>\n<p>\u6307\u7b97\u6cd5\u5360\u7528\u7684\u5185\u5b58\u5c11&#xff0c;\u5373\u7a7a\u95f4\u590d\u6742\u5ea6\u4f4e\u3002<\/p>\n<p>&#x1f4a1; \u8bb0\u5fc6\u53e3\u8bc0&#xff1a;\u201c\u6b63\u53ef\u8bfb\u5199\u5065&#xff0c;\u9ad8\u6548\u4f4e\u5b58\u50a8\u201d\u2014\u2014\u6b63\u786e\u6027\u3001\u53ef\u4f7f\u7528\u6027\u3001\u53ef\u8bfb\u6027\u3001\u5065\u58ee\u6027\u3001\u9ad8\u6548\u7387\u3001\u4f4e\u5b58\u50a8\u91cf\u9700\u6c42\u3002<\/p>\n<p>\u26a0\ufe0f \u6ce8\u610f\u4e00\u4e2a\u5e38\u89c1\u7684\u8003\u8bd5\u9677\u9631&#xff1a;\u9ad8\u6548\u7387\u4e0e\u4f4e\u5b58\u50a8\u91cf\u5f80\u5f80\u4e0d\u53ef\u517c\u5f97&#xff08;\u5178\u578b\u7684&#034;\u7a7a\u95f4\u6362\u65f6\u95f4&#034;\u6216&#034;\u65f6\u95f4\u6362\u7a7a\u95f4&#034;\u7684\u6743\u8861&#xff0c;trade-off&#xff09;\u3002\u4f8b\u5982\u54c8\u5e0c\u8868\u67e5\u627e\u662f O(1) \u65f6\u95f4&#xff0c;\u4f46\u9700\u8981\u989d\u5916 O(n) \u7a7a\u95f4&#xff1b;\u800c\u76f4\u63a5\u904d\u5386\u662f O(1) \u7a7a\u95f4&#xff0c;\u4f46\u9700\u8981 O(n) \u65f6\u95f4\u3002\u7b97\u6cd5\u8bbe\u8ba1\u7684\u827a\u672f&#xff0c;\u5f88\u5927\u7a0b\u5ea6\u4e0a\u5c31\u662f\u5728\u8fd9\u4e9b\u76ee\u6807\u4e4b\u95f4\u505a\u53d6\u820d\u3002<\/p>\n<hr \/>\n<h3>\u56db\u3001\u3010\u4f8b1-1\u3011\u5355\u94fe\u8868\u67e5\u627e\u7b97\u6cd5\u7684\u9519\u8bef\u5206\u6790&#xff08;\u7ecf\u5178\u4f8b\u9898&#xff09;<\/h3>\n<p>\u8fd9\u9053\u4f8b\u9898\u662f&#034;\u516d\u5927\u76ee\u6807&#034;\u4e2d\u6b63\u786e\u6027\u4e0e\u5065\u58ee\u6027\u7684\u6d3b\u6559\u6750&#xff0c;\u51e0\u4e4e\u6bcf\u5e74\u90fd\u4f1a\u88ab\u62ff\u6765\u5f53\u5206\u6790\u9898\u8003&#xff0c;\u52a1\u5fc5\u5403\u900f\u3002<\/p>\n<h4>4.1 \u9898\u76ee\u63cf\u8ff0<\/h4>\n<p>\u4ee5\u4e0b\u7b97\u6cd5\u7528\u4e8e\u5728\u5e26\u5934\u7ed3\u70b9\u7684\u5355\u94fe\u8868 h \u4e2d\u67e5\u627e\u7b2c\u4e00\u4e2a\u503c\u4e3a x \u7684\u7ed3\u70b9&#xff0c;\u627e\u5230\u540e\u8fd4\u56de\u5176\u903b\u8f91\u5e8f\u53f7&#xff08;\u4ece1\u8ba1\u8d77&#xff09;&#xff0c;\u5426\u5219\u8fd4\u56de0\u3002\u5206\u6790\u8be5\u7b97\u6cd5\u5b58\u5728\u7684\u95ee\u9898\u3002<\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">findx<\/span><span class=\"token punctuation\">(<\/span>LNode <span class=\"token operator\">*<\/span>h<span class=\"token punctuation\">;<\/span> <span class=\"token keyword\">int<\/span> x<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    LNode <span class=\"token operator\">*<\/span>p <span class=\"token operator\">&#061;<\/span> h<span class=\"token operator\">-&gt;<\/span>next<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>p<span class=\"token operator\">-&gt;<\/span>data <span class=\"token operator\">!&#061;<\/span> x<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        p <span class=\"token operator\">&#061;<\/span> p<span class=\"token operator\">-&gt;<\/span>next<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>4.2 \u95ee\u9898\u8bca\u65ad<\/h4>\n<p>\u7ed3\u8bba\u5148\u884c&#xff1a;\u8be5\u7b97\u6cd5\u4e0d\u6ee1\u8db3\u6b63\u786e\u6027\u548c\u5065\u58ee\u6027\u3002 \u5177\u4f53\u6709\u4e24\u4e2a\u81f4\u547d\u95ee\u9898&#xff1a;<\/p>\n<p>\u95ee\u9898&#xff08;1&#xff09;&#xff1a;\u521d\u503c\u9519\u8bef&#xff0c;\u8fdd\u53cd\u6b63\u786e\u6027<\/p>\n<p>\u5f53\u5355\u94fe\u8868\u4e2d\u9996\u7ed3\u70b9&#xff08;\u7b2c\u4e00\u4e2a\u6570\u636e\u7ed3\u70b9&#xff09;\u7684\u503c\u5c31\u662f x \u65f6&#xff1a;<\/p>\n<ul>\n<li>\u8fdb\u5165\u5faa\u73af\u524d&#xff0c;p \u6307\u5411\u9996\u7ed3\u70b9&#xff0c;i &#061; 0&#xff1b;<\/li>\n<li>\u5faa\u73af\u6761\u4ef6 p-&gt;data !&#061; x \u4e0d\u6210\u7acb&#xff0c;\u76f4\u63a5\u8df3\u8fc7\u5faa\u73af\u4f53&#xff1b;<\/li>\n<li>return i&#xff0c;\u8fd4\u56de 0\u3002<\/li>\n<\/ul>\n<p>\u4f46\u6309\u7167\u9898\u610f&#xff0c;\u9996\u7ed3\u70b9\u7684\u903b\u8f91\u5e8f\u53f7\u5e94\u5f53\u662f 1&#xff01;\u8fd4\u56de\u503c\u9519\u8bef&#xff0c;\u8fd9\u5c31\u662f\u6b63\u786e\u6027\u95ee\u9898\u3002<\/p>\n<p>\u95ee\u9898&#xff08;2&#xff09;&#xff1a;\u5faa\u73af\u6761\u4ef6\u7f3a\u5c11\u5224\u7a7a&#xff0c;\u8fdd\u53cd\u5065\u58ee\u6027<\/p>\n<p>\u5f53\u5355\u94fe\u8868\u4e2d\u4e0d\u5b58\u5728\u503c\u4e3a x \u7684\u7ed3\u70b9\u65f6&#xff1a;<\/p>\n<ul>\n<li>\u5faa\u73af\u4f1a\u4e00\u8def\u8d70\u5230\u5e95&#xff0c;p \u6700\u7ec8\u53d8\u6210 NULL&#xff1b;<\/li>\n<li>\u4f46\u5faa\u73af\u6761\u4ef6 p-&gt;data !&#061; x \u4ecd\u7136\u4f1a\u53bb\u89e3\u5f15\u7528 p&#xff0c;\u4e5f\u5c31\u662f\u5bf9\u7a7a\u6307\u9488\u6267\u884c p-&gt;data \u548c p &#061; p-&gt;next&#xff1b;<\/li>\n<li>\u7ed3\u679c\u662f\u975e\u6cd5\u5185\u5b58\u8bbf\u95ee&#xff0c;\u7a0b\u5e8f\u76f4\u63a5\u5d29\u6e83\u3002<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f\u5065\u58ee\u6027\u95ee\u9898\u2014\u2014\u7b97\u6cd5\u6ca1\u6709\u5bf9&#034;\u67e5\u4e0d\u5230&#034;\u8fd9\u79cd\u5b8c\u5168\u53ef\u80fd\u53d1\u751f\u7684\u60c5\u51b5\u505a\u51fa\u6070\u5f53\u5904\u7406\u3002<\/p>\n<h4>4.3 \u6539\u6b63\u540e\u7684\u4ee3\u7801<\/h4>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">findx<\/span><span class=\"token punctuation\">(<\/span>LNode <span class=\"token operator\">*<\/span>h<span class=\"token punctuation\">;<\/span> <span class=\"token keyword\">int<\/span> x<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    LNode <span class=\"token operator\">*<\/span>p <span class=\"token operator\">&#061;<\/span> h<span class=\"token operator\">-&gt;<\/span>next<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span>                              <span class=\"token comment\">\/\/ \u2460 \u521d\u503c\u6539\u4e3a1<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>p <span class=\"token operator\">!&#061;<\/span> <span class=\"token constant\">NULL<\/span> <span class=\"token operator\">&amp;&amp;<\/span> p<span class=\"token operator\">-&gt;<\/span>data <span class=\"token operator\">!&#061;<\/span> x<span class=\"token punctuation\">)<\/span>       <span class=\"token comment\">\/\/ \u2461 \u5faa\u73af\u6761\u4ef6\u5148\u5224\u7a7a<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        p <span class=\"token operator\">&#061;<\/span> p<span class=\"token operator\">-&gt;<\/span>next<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>p <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token constant\">NULL<\/span><span class=\"token punctuation\">)<\/span>                          <span class=\"token comment\">\/\/ \u2462 \u533a\u5206\u67e5\u627e\u6210\u529f\/\u5931\u8d25<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">else<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>4.4 \u9519\u8bef\u5206\u6790\u8981\u70b9&#xff08;\u8003\u8bd5\u7b54\u9898\u6a21\u677f&#xff09;<\/h4>\n<li>\u521d\u503c\u9519\u8bef&#xff1a;i \u521d\u59cb\u5199\u6210 0\u3002\u7b2c\u4e00\u4e2a\u6709\u6548\u7ed3\u70b9\u7684\u903b\u8f91\u5e8f\u53f7\u662f1&#xff0c;i \u5e94\u5f53\u521d\u59cb\u5316\u4e3a 1\u3002\u8fd9\u6837\u82e5\u7b2c\u4e00\u4e2a\u7ed3\u70b9\u5c31\u662f x&#xff0c;\u5faa\u73af\u4e0d\u6267\u884c&#xff0c;\u76f4\u63a5\u8fd4\u56de i &#061; 1&#xff0c;\u5f97\u5230\u6b63\u786e\u5e8f\u53f7\u3002<\/li>\n<li>\u5faa\u73af\u6761\u4ef6\u7f3a\u5c11\u5224\u7a7a&#xff1a;\u539f\u6765\u5199\u7684\u662f while(p-&gt;data !&#061; x)&#xff0c;\u6ca1\u6709\u5224\u65ad p !&#061; NULL\u3002\u94fe\u8868\u627e\u4e0d\u5230 x \u65f6&#xff0c;p \u4f1a\u53d8\u6210 NULL&#xff0c;\u518d\u8bbf\u95ee p-&gt;data \u4f1a\u9020\u6210\u7a7a\u6307\u9488\u8bbf\u95ee&#xff0c;\u7a0b\u5e8f\u5d29\u6e83\u3002\u5faa\u73af\u6761\u4ef6\u5fc5\u987b\u5199\u6210 p !&#061; NULL &amp;&amp; p-&gt;data !&#061; x&#xff0c;\u5229\u7528 &amp;&amp; \u7684\u77ed\u8def\u6c42\u503c\u7279\u6027&#xff1a;\u4e00\u65e6 p &#061;&#061; NULL&#xff0c;\u540e\u534a\u90e8\u5206\u4e0d\u4f1a\u518d\u6267\u884c\u3002<\/li>\n<li>\u6ca1\u6709\u533a\u5206\u67e5\u627e\u6210\u529f\/\u5931\u8d25&#xff1a;\u539f\u4ee3\u7801\u65e0\u8bba\u5982\u4f55\u90fd\u76f4\u63a5 return i\u3002\u6b63\u786e\u505a\u6cd5\u662f\u5faa\u73af\u7ed3\u675f\u540e\u5224\u65ad&#xff1a;p &#061;&#061; NULL \u8bf4\u660e\u904d\u5386\u5b8c\u6574\u4e2a\u94fe\u8868\u90fd\u6ca1\u627e\u5230&#xff0c;\u5e94\u8fd4\u56de 0&#xff1b;\u5426\u5219\u8bf4\u660e\u627e\u5230\u4e86&#xff0c;\u8fd4\u56de\u5f53\u524d\u5e8f\u53f7 i\u3002<\/li>\n<h4>4.5 \u4e3e\u4e00\u53cd\u4e09&#xff1a;\u8fd9\u7c7b\u9898\u7684\u901a\u7528\u68c0\u67e5\u6e05\u5355<\/h4>\n<p>\u4ee5\u540e\u51e1\u662f\u62ff\u5230\u4e00\u6bb5&#034;\u627e\u832c&#034;\u4ee3\u7801&#xff0c;\u53ef\u4ee5\u6309\u4ee5\u4e0b\u6e05\u5355\u9010\u9879\u6392\u67e5&#xff1a;<\/p>\n<ul>\n<li>\u2705 \u5faa\u73af\u53d8\u91cf\u7684\u521d\u503c\u5bf9\u4e0d\u5bf9&#xff1f;&#xff08;\u5dee\u4e00\u9519\u8bef&#xff0c;off-by-one&#xff09;<\/li>\n<li>\u2705 \u5faa\u73af\u6761\u4ef6\u6709\u6ca1\u6709\u8fb9\u754c\u4fdd\u62a4&#xff08;\u5224\u7a7a\u3001\u5224\u8d8a\u754c&#xff09;&#xff1f;<\/li>\n<li>\u2705 \u77ed\u8def\u6c42\u503c\u7684\u987a\u5e8f\u5bf9\u4e0d\u5bf9&#xff1f;&#xff08;\u5fc5\u987b\u5148\u5224 p !&#061; NULL \u518d\u5224 p-&gt;data&#xff0c;\u987a\u5e8f\u53cd\u4e86\u7167\u6837\u5d29&#xff09;<\/li>\n<li>\u2705 \u6210\u529f\u548c\u5931\u8d25\u4e24\u79cd\u8def\u5f84\u7684\u8fd4\u56de\u503c\u662f\u5426\u90fd\u7b26\u5408\u9898\u610f&#xff1f;<\/li>\n<li>\u2705 \u5e26\u5934\u7ed3\u70b9\u7684\u94fe\u8868&#xff0c;p \u5e94\u8be5\u4ece h-&gt;next \u5f00\u59cb&#xff08;\u5934\u7ed3\u70b9\u4e0d\u5b58\u6570\u636e&#xff0c;\u5e8f\u53f7\u4ece1\u8ba1&#xff09;\u2014\u2014\u8fd9\u4e00\u70b9\u539f\u4ee3\u7801\u662f\u5bf9\u7684&#xff0c;\u522b\u8bef\u4f24\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u4e94\u3001\u7b97\u6cd5\u7684\u4e94\u5927\u7279\u5f81&#xff08;Knuth\u5b9a\u4e49&#xff09;<\/h3>\n<p>\u7b97\u6cd5\u8bbe\u8ba1\u7684\u5148\u9a71\u8005\u5510\u7eb3\u5fb7\u00b7E\u00b7\u514b\u52aa\u7279&#xff08;Donald E. Knuth&#xff09;\u2014\u2014\u300a\u8ba1\u7b97\u673a\u7a0b\u5e8f\u8bbe\u8ba1\u827a\u672f\u300b(TAOCP) \u7684\u4f5c\u8005\u3001\u56fe\u7075\u5956\u5f97\u4e3b\u3001TeX\u6392\u7248\u7cfb\u7edf\u7684\u521b\u9020\u8005\u2014\u2014\u7ed9\u51fa\u4e86\u7b97\u6cd5\u7684\u4e94\u4e2a\u7ecf\u5178\u7279\u5f81&#xff1a;<\/p>\n<h4>5.1 \u6709\u7a77\u6027&#xff08;Finiteness&#xff09;<\/h4>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u6240\u5305\u542b\u7684\u8ba1\u7b97\u6b65\u9aa4\u662f\u6709\u9650\u7684\u3002<\/p>\n<p>\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u7b97\u6cd5\u5fc5\u987b\u5728\u6267\u884c\u6709\u9650\u6b65\u4e4b\u540e\u7ed3\u675f&#xff0c;\u4e0d\u80fd\u9677\u5165\u6c38\u65e0\u4f11\u6b62\u7684\u6b7b\u5faa\u73af\u3002\u6ce8\u610f&#xff1a;\u7a0b\u5e8f\u4e0d\u4e00\u5b9a\u6709\u7a77&#xff08;\u6bd4\u5982\u64cd\u4f5c\u7cfb\u7edf\u662f\u4e2a\u65e0\u9650\u5faa\u73af\u7684\u7a0b\u5e8f&#xff09;&#xff0c;\u4f46\u7b97\u6cd5\u5fc5\u987b\u6709\u7a77\u3002\u8fd9\u662f\u7b97\u6cd5\u4e0e\u7a0b\u5e8f\u7684\u91cd\u8981\u533a\u522b\u4e4b\u4e00\u3002<\/p>\n<h4>5.2 \u786e\u5b9a\u6027&#xff08;Definiteness&#xff09;<\/h4>\n<p>\u7b97\u6cd5\u6267\u884c\u7684\u6bcf\u4e00\u4e2a\u6b65\u9aa4\u5fc5\u987b\u6709\u786e\u5207\u7684\u5b9a\u4e49&#xff0c;\u4e0d\u80fd\u6709\u6a21\u68f1\u4e24\u53ef\u7684\u60c5\u51b5\u3002<\/p>\n<p>\u4f8b\u5982&#034;\u628a x \u52a0\u4e0a\u4e00\u4e2a\u8db3\u591f\u5927\u7684\u6570&#034;\u5c31\u4e0d\u662f\u786e\u5b9a\u7684\u63cf\u8ff0\u2014\u2014\u591a\u5927\u7b97&#034;\u8db3\u591f\u5927&#034;&#xff1f;\u800c&#034;x &#061; x &#043; 5&#034;\u662f\u786e\u5b9a\u7684\u3002\u786e\u5b9a\u6027\u8981\u6c42\u6bcf\u6761\u6307\u4ee4\u90fd\u6709\u552f\u4e00\u3001\u6e05\u6670\u7684\u542b\u4e49&#xff0c;\u540c\u6837\u7684\u8f93\u5165\u6c38\u8fdc\u8d70\u540c\u6837\u7684\u8def\u5f84\u3001\u5f97\u540c\u6837\u7684\u7ed3\u679c\u3002<\/p>\n<h4>5.3 \u6570\u636e\u8f93\u5165&#xff08;Input&#xff09;<\/h4>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u6709\u96f6\u4e2a\u6216\u591a\u4e2a\u6570\u636e\u8f93\u5165\u3002<\/p>\n<p>\u6ce8\u610f\u662f&#034;\u96f6\u4e2a\u6216&#034;\u2014\u2014\u6709\u7684\u7b97\u6cd5\u4e0d\u9700\u8981\u5916\u90e8\u8f93\u5165&#xff08;\u6bd4\u5982\u6253\u5370\u524d100\u4e2a\u7d20\u6570&#xff09;&#xff0c;\u8f93\u5165\u53ef\u4ee5\u5185\u5d4c\u5728\u7b97\u6cd5\u4e4b\u4e2d\u3002<\/p>\n<h4>5.4 \u6570\u636e\u8f93\u51fa&#xff08;Output&#xff09;<\/h4>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u6709\u4e00\u4e2a\u6216\u591a\u4e2a\u6570\u636e\u8f93\u51fa&#xff0c;\u6ca1\u6709\u8f93\u51fa\u7684\u7b97\u6cd5\u662f\u6ca1\u6709\u610f\u4e49\u7684\u3002<\/p>\n<p>\u8f93\u5165\u53ef\u4ee5\u662f\u96f6\u4e2a&#xff0c;\u4f46\u8f93\u51fa\u81f3\u5c11\u4e00\u4e2a\u3002\u7b97\u6cd5\u7684\u76ee\u7684\u5c31\u662f\u52a0\u5de5\u6570\u636e\u4ea7\u51fa\u7ed3\u679c&#xff0c;\u4ec0\u4e48\u90fd\u4e0d\u8f93\u51fa\u7684&#034;\u7b97\u6cd5&#034;\u505a\u4e86\u65e0\u7528\u529f\u3002<\/p>\n<h4>5.5 \u53ef\u884c\u6027&#xff08;Effectiveness \/ Feasibility&#xff09;<\/h4>\n<p>\u6bcf\u4e2a\u6b65\u9aa4\u90fd\u53ef\u4ee5\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u3002<\/p>\n<p>\u4e5f\u7ffb\u8bd1\u4e3a&#034;\u6709\u6548\u6027&#034;\u3002\u7b97\u6cd5\u4e2d\u7684\u6bcf\u4e00\u4e2a\u64cd\u4f5c\u90fd\u5fc5\u987b\u662f\u53ef\u6267\u884c\u7684\u57fa\u672c\u64cd\u4f5c&#xff0c;\u80fd\u591f\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u3002\u9664\u4ee5\u96f6\u3001\u5f00\u8d1f\u6570\u7684\u5b9e\u6570\u5e73\u65b9\u6839\u3001\u8c03\u7528\u4e00\u4e2a\u4e0d\u5b58\u5728\u7684\u786c\u4ef6\u529f\u80fd\u2014\u2014\u8fd9\u4e9b\u90fd\u662f\u4e0d\u53ef\u884c\u7684\u64cd\u4f5c\u3002<\/p>\n<h4>5.6 \u901f\u8bb0\u603b\u7ed3<\/h4>\n<table>\n<tr>\u7279\u5f81\u4e00\u53e5\u8bdd\u8bb0\u5fc6\u5173\u952e\u8bcd<\/tr>\n<tbody>\n<tr>\n<td>\u2705 \u6709\u7a77\u6027<\/td>\n<td>\u6b65\u9aa4\u6267\u884c\u6709\u9650\u6b21\u540e\u5fc5\u987b\u7ed3\u675f<\/td>\n<td>\u4e0d\u80fd\u6b7b\u5faa\u73af<\/td>\n<\/tr>\n<tr>\n<td>\u2705 \u786e\u5b9a\u6027<\/td>\n<td>\u6bcf\u4e00\u6b65\u5b9a\u4e49\u660e\u786e&#xff0c;\u65e0\u6b67\u4e49<\/td>\n<td>\u4e0d\u80fd\u6a21\u68f1\u4e24\u53ef<\/td>\n<\/tr>\n<tr>\n<td>\u2705 \u8f93\u5165<\/td>\n<td>0\u4e2a\u6216\u591a\u4e2a\u8f93\u5165<\/td>\n<td>\u53ef\u4ee5\u4e3a\u96f6<\/td>\n<\/tr>\n<tr>\n<td>\u2705 \u8f93\u51fa<\/td>\n<td>1\u4e2a\u6216\u591a\u4e2a\u8f93\u51fa<\/td>\n<td>\u81f3\u5c11\u4e00\u4e2a<\/td>\n<\/tr>\n<tr>\n<td>\u2705 \u53ef\u884c\u6027<\/td>\n<td>\u6bcf\u4e2a\u64cd\u4f5c\u90fd\u80fd\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210<\/td>\n<td>\u4e0d\u80fd\u9664\u96f6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u26a0\ufe0f \u9ad8\u9891\u8003\u70b9\u8fa8\u6790&#xff1a;\u6709\u7a77\u6027 vs \u53ef\u884c\u6027\u600e\u4e48\u533a\u5206&#xff1f;<\/p>\n<ul>\n<li>\u6709\u7a77\u6027\u5173\u6ce8\u7684\u662f\u6574\u4f53&#xff1a;\u6574\u4e2a\u7b97\u6cd5\u7684\u6b65\u9aa4\u603b\u6570\u6709\u9650&#xff0c;\u6700\u7ec8\u4f1a\u505c\u4e0b\u6765&#xff1b;<\/li>\n<li>\u53ef\u884c\u6027\u5173\u6ce8\u7684\u662f\u5355\u6b65&#xff1a;\u6bcf\u4e00\u6b65\u64cd\u4f5c\u672c\u8eab\u662f\u53ef\u6267\u884c\u7684\u3001\u80fd\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u7684\u3002<\/li>\n<li>\u6b7b\u5faa\u73af \u2192 \u8fdd\u53cd\u6709\u7a77\u6027&#xff1b;\u9664\u4ee5\u96f6\u8fd9\u79cd\u6839\u672c\u6267\u884c\u4e0d\u4e86\u7684\u64cd\u4f5c \u2192 \u8fdd\u53cd\u53ef\u884c\u6027\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u516d\u3001\u6848\u4f8b&#xff1a;\u6c42 6x&#043;5y&#043;4z&#061;50 \u7684\u6b63\u6574\u6570\u89e3<\/h3>\n<p>\u4e0b\u9762\u7528\u4e00\u4e2a\u5177\u4f53\u7684\u6570\u5b66\u95ee\u9898&#xff0c;\u4f53\u4f1a\u7528\u81ea\u7136\u8bed\u8a00&#xff08;\u6b65\u9aa4\u5316\u63cf\u8ff0&#xff09;\u6765\u8868\u8fbe\u4e00\u4e2a\u7b97\u6cd5\u662f\u4ec0\u4e48\u611f\u89c9\u3002<\/p>\n<h4>6.1 \u95ee\u9898<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          6<\/p>\n<p>          x<\/p>\n<p>          &#043;<\/p>\n<p>          5<\/p>\n<p>          y<\/p>\n<p>          &#043;<\/p>\n<p>          4<\/p>\n<p>          z<\/p>\n<p>          &#061;<\/p>\n<p>          50<\/p>\n<p>       \\\\boldsymbol{6x&#043;5y&#043;4z&#061;50}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">6<\/span><span class=\"mord boldsymbol\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">5<\/span><span class=\"mord boldsymbol\" style=\"margin-right: 0.037em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">4<\/span><span class=\"mord boldsymbol\" style=\"margin-right: 0.0421em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">50<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6709\u591a\u5c11\u7ec4\u6b63\u6574\u6570\u89e3&#xff1f;<\/p>\n<h4>6.2 \u6570\u5b66\u5206\u6790&#xff1a;\u786e\u5b9a\u679a\u4e3e\u8fb9\u754c<\/h4>\n<p>\u7531\u4e8e x, y, z \u90fd\u662f\u6b63\u6574\u6570&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        x<\/p>\n<p>        ,<\/p>\n<p>        y<\/p>\n<p>        ,<\/p>\n<p>        z<\/p>\n<p>        \u2265<\/p>\n<p>        1<\/p>\n<p>       x, y, z \\\\ge 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8304em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u4e14\u5404\u9879\u90fd\u662f\u6b63\u6570&#xff0c;\u6240\u4ee5\u6bcf\u4e00\u9879\u90fd\u5fc5\u987b\u5c0f\u4e8e50&#xff1a;<\/p>\n<ul>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         6<\/p>\n<p>         x<\/p>\n<p>         &lt;<\/p>\n<p>         50<\/p>\n<p>         \u21d2<\/p>\n<p>         x<\/p>\n<p>         \u2264<\/p>\n<p>         8<\/p>\n<p>        6x &lt; 50 \\\\Rightarrow x \\\\le 8<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6835em;vertical-align: -0.0391em\"><\/span><span class=\"mord\">6<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">50<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4f46\u56e0\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         5<\/p>\n<p>         y<\/p>\n<p>         &#043;<\/p>\n<p>         4<\/p>\n<p>         z<\/p>\n<p>         \u2265<\/p>\n<p>         5<\/p>\n<p>         &#043;<\/p>\n<p>         4<\/p>\n<p>         &#061;<\/p>\n<p>         9<\/p>\n<p>        5y&#043;4z \\\\ge 5&#043;4&#061;9<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">5<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">4<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5b9e\u9645\u4e0a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         6<\/p>\n<p>         x<\/p>\n<p>         \u2264<\/p>\n<p>         41<\/p>\n<p>        6x \\\\le 41<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">6<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">41<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>         \u2264<\/p>\n<p>         6<\/p>\n<p>        x \\\\le 6<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">6<\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li>\u540c\u7406 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         5<\/p>\n<p>         y<\/p>\n<p>         &lt;<\/p>\n<p>         50<\/p>\n<p>         \u21d2<\/p>\n<p>         y<\/p>\n<p>         \u2264<\/p>\n<p>         9<\/p>\n<p>        5y &lt; 50 \\\\Rightarrow y \\\\le 9<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">5<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">50<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8304em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u8003\u8651 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         6<\/p>\n<p>         x<\/p>\n<p>         &#043;<\/p>\n<p>         4<\/p>\n<p>         z<\/p>\n<p>         \u2265<\/p>\n<p>         10<\/p>\n<p>        6x&#043;4z \\\\ge 10<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">6<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">10<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         5<\/p>\n<p>         y<\/p>\n<p>         \u2264<\/p>\n<p>         40<\/p>\n<p>        5y \\\\le 40<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">5<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">40<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         y<\/p>\n<p>         \u2264<\/p>\n<p>         8<\/p>\n<p>        y \\\\le 8<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8304em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         4<\/p>\n<p>         z<\/p>\n<p>         &lt;<\/p>\n<p>         50<\/p>\n<p>         \u21d2<\/p>\n<p>         z<\/p>\n<p>         \u2264<\/p>\n<p>         12<\/p>\n<p>        4z &lt; 50 \\\\Rightarrow z \\\\le 12<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6835em;vertical-align: -0.0391em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">50<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">12<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u8003\u8651 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         6<\/p>\n<p>         x<\/p>\n<p>         &#043;<\/p>\n<p>         5<\/p>\n<p>         y<\/p>\n<p>         \u2265<\/p>\n<p>         11<\/p>\n<p>        6x&#043;5y \\\\ge 11<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">6<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">5<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">11<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         4<\/p>\n<p>         z<\/p>\n<p>         \u2264<\/p>\n<p>         39<\/p>\n<p>        4z \\\\le 39<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">39<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         z<\/p>\n<p>         \u2264<\/p>\n<p>         9<\/p>\n<p>        z \\\\le 9<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<\/ul>\n<p>\u6559\u6750\u91c7\u7528\u7684\u8fb9\u754c\u4e3a&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         6<\/p>\n<p>         x<\/p>\n<p>         &lt;<\/p>\n<p>         50<\/p>\n<p>         \u21d2<\/p>\n<p>         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class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">6<\/span><\/span><\/span><\/span><\/span>&#xff1b;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         5<\/p>\n<p>         y<\/p>\n<p>         &lt;<\/p>\n<p>         50<\/p>\n<p>         \u21d2<\/p>\n<p>         y<\/p>\n<p>         \u2264<\/p>\n<p>         8<\/p>\n<p>        5y&lt;50 \\\\Rightarrow y\\\\le8<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">5<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">50<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8304em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span><\/span>&#xff1b;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         4<\/p>\n<p>         z<\/p>\n<p>         &lt;<\/p>\n<p>         50<\/p>\n<p>         \u21d2<\/p>\n<p>         z<\/p>\n<p>         \u2264<\/p>\n<p>         9<\/p>\n<p>        4z&lt;50 \\\\Rightarrow z\\\\le9<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6835em;vertical-align: -0.0391em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">50<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.044em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span><\/span>\u3002&#xff08;\u8fd9\u91cc\u5df2\u7ecf\u9690\u542b\u6263\u9664\u4e86\u53e6\u5916\u4e24\u9879\u7684\u6700\u5c0f\u53d6\u503c&#xff0c;\u4fdd\u8bc1\u4e0d\u9057\u6f0f\u89e3\u3002&#xff09;<\/p>\n<h4>6.3 \u7b97\u6cd5\u7684\u81ea\u7136\u8bed\u8a00\u63cf\u8ff0&#xff08;\u7a77\u4e3e\u6cd5&#xff09;<\/h4>\n<p>\u2460 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          t<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{t&#061;0;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">t<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">0<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> &#xff08;t \u7528\u4e8e\u7edf\u8ba1\u89e3\u7684\u4e2a\u6570&#xff09; \u2461 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          x<\/p>\n<p>          &#061;<\/p>\n<p>          1<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{x&#061;1;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">1<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u2462 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          y<\/p>\n<p>          &#061;<\/p>\n<p>          1<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{y&#061;1;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.037em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">1<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u2463 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          z<\/p>\n<p>          &#061;<\/p>\n<p>          1<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{z&#061;1;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.0421em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">1<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u2464 \u5982\u679c\u6ee1\u8db3\u5f0f\u5b50 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          6<\/p>\n<p>          x<\/p>\n<p>          &#043;<\/p>\n<p>          5<\/p>\n<p>       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boldsymbol\">t<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u5e76\u8f93\u51fa\u8fd9\u4e2a\u89e3&#xff08;\u8f93\u51fa x, y, z \u7684\u503c&#xff09;&#xff1b; \u2465 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          z<\/p>\n<p>          &#061;<\/p>\n<p>          z<\/p>\n<p>          &#043;<\/p>\n<p>          1<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{z&#061;z&#043;1;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.0421em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord boldsymbol\" style=\"margin-right: 0.0421em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">1<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u2466 \u5982\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          z<\/p>\n<p>          \u2264<\/p>\n<p>          9<\/p>\n<p>       \\\\boldsymbol{z \\\\le 9}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8933em;vertical-align: -0.1967em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.0421em\">z<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">9<\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219\u8f6c\u6b65\u9aa4\u2464&#xff0c;\u5426\u5219\u7ee7\u7eed\u2467&#xff1b; \u2467 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          y<\/p>\n<p>          &#061;<\/p>\n<p>          y<\/p>\n<p>          &#043;<\/p>\n<p>          1<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{y&#061;y&#043;1;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.037em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord boldsymbol\" style=\"margin-right: 0.037em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">1<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u2468 \u5982\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          y<\/p>\n<p>          \u2264<\/p>\n<p>          8<\/p>\n<p>       \\\\boldsymbol{y \\\\le 8}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8933em;vertical-align: -0.1967em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.037em\">y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">8<\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219\u8f6c\u6b65\u9aa4\u2463&#xff0c;\u5426\u5219\u7ee7\u7eed\u2469&#xff1b; \u2469 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          x<\/p>\n<p>          &#061;<\/p>\n<p>          x<\/p>\n<p>          &#043;<\/p>\n<p>          1<\/p>\n<p>          ;<\/p>\n<p>       \\\\boldsymbol{x&#061;x&#043;1;}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord boldsymbol\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">1<\/span><span class=\"mpunct mathbf\">;<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u246a \u5982\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          x<\/p>\n<p>          \u2264<\/p>\n<p>          6<\/p>\n<p>       \\\\boldsymbol{x \\\\le 6}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8933em;vertical-align: -0.1967em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel mathbf\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord mathbf\">6<\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219\u8f6c\u6b65\u9aa4\u2462&#xff0c;\u5426\u5219\u7ee7\u7eed\u246b&#xff1b; \u246b \u7ed3\u675f\u3002<\/p>\n<h4>6.4 \u7b97\u6cd5\u7ed3\u6784\u89e3\u8bfb<\/h4>\n<p>\u8fd9\u6bb5\u81ea\u7136\u8bed\u8a00\u63cf\u8ff0\u672c\u8d28\u4e0a\u662f\u4e00\u4e2a\u4e09\u91cd\u5d4c\u5957\u5faa\u73af\u7684\u7a77\u4e3e&#xff08;\u66b4\u529b\u679a\u4e3e&#xff09;\u7b97\u6cd5&#xff1a;<\/p>\n<ul>\n<li>\u6700\u5916\u5c42\u5faa\u73af\u679a\u4e3e x&#xff08;1\u21926&#xff09;&#xff0c;\u4e2d\u5c42\u5faa\u73af\u679a\u4e3e y&#xff08;1\u21928&#xff09;&#xff0c;\u6700\u5185\u5c42\u5faa\u73af\u679a\u4e3e z&#xff08;1\u21929&#xff09;&#xff1b;<\/li>\n<li>\u6bcf\u6362\u4e00\u4e2a y&#xff0c;z \u8981\u91cd\u65b0\u4ece1\u5f00\u59cb&#xff08;\u6b65\u9aa4\u2468\u8f6c\u56de\u6b65\u9aa4\u2463&#xff09;&#xff1b;\u6bcf\u6362\u4e00\u4e2a x&#xff0c;y \u8981\u91cd\u65b0\u4ece1\u5f00\u59cb&#xff08;\u6b65\u9aa4\u246a\u8f6c\u56de\u6b65\u9aa4\u2462&#xff09;\u2014\u2014\u8fd9\u6b63\u662f\u5d4c\u5957\u5faa\u73af&#034;\u5185\u5c42\u91cd\u7f6e&#034;\u7684\u8bed\u4e49&#xff1b;<\/li>\n<li>\u5185\u5c42\u6bcf\u679a\u4e3e\u4e00\u7ec4 (x, y, z)&#xff0c;\u5c31\u9a8c\u8bc1\u4e00\u6b21\u7b49\u5f0f&#xff0c;\u6210\u7acb\u5219\u8ba1\u6570\u5e76\u8f93\u51fa\u3002<\/li>\n<\/ul>\n<p>\u5982\u679c\u7ffb\u8bd1\u6210 C \u8bed\u8a00&#xff0c;\u5c31\u662f&#xff1a;<\/p>\n<p><span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">include<\/span> <span class=\"token string\">&lt;stdio.h&gt;<\/span><\/span><\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">main<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> t <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span>                        <span class=\"token comment\">\/\/ \u89e3\u7684\u4e2a\u6570<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> x <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> x <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">6<\/span><span class=\"token punctuation\">;<\/span> x<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span>      <span class=\"token comment\">\/\/ \u679a\u4e3ex<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> y <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> y <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">8<\/span><span class=\"token punctuation\">;<\/span> y<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\">\/\/ \u679a\u4e3ey<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> z <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> z <span class=\"token operator\">&lt;&#061;<\/span> <span class=\"token number\">9<\/span><span class=\"token punctuation\">;<\/span> z<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span>  <span class=\"token comment\">\/\/ \u679a\u4e3ez<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token number\">6<\/span><span class=\"token operator\">*<\/span>x <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">5<\/span><span class=\"token operator\">*<\/span>y <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">4<\/span><span class=\"token operator\">*<\/span>z <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">50<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token punctuation\">{<\/span><br \/>\n                    t<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                    <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;x&#061;%d, y&#061;%d, z&#061;%d\\\\n&#034;<\/span><span class=\"token punctuation\">,<\/span> x<span class=\"token punctuation\">,<\/span> y<span class=\"token punctuation\">,<\/span> z<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n                <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5171\u6709%d\u7ec4\u6b63\u6574\u6570\u89e3\\\\n&#034;<\/span><span class=\"token punctuation\">,<\/span> t<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>6.5 \u601d\u8003&#xff1a;\u8fd9\u4e2a\u63cf\u8ff0\u7b26\u5408\u7b97\u6cd5\u4e94\u5927\u7279\u5f81\u5417&#xff1f;<\/h4>\n<ul>\n<li>\u6709\u7a77\u6027 \u2705&#xff1a;x \u6700\u591a6\u79cd\u3001y \u6700\u591a8\u79cd\u3001z \u6700\u591a9\u79cd\u53d6\u503c&#xff0c;\u603b\u9a8c\u8bc1\u6b21\u6570\u4e0d\u8d85\u8fc7 6\u00d78\u00d79 &#061; 432 \u6b21&#xff0c;\u5fc5\u7136\u7ed3\u675f&#xff1b;<\/li>\n<li>\u786e\u5b9a\u6027 \u2705&#xff1a;\u6bcf\u4e00\u6b65&#xff08;\u8d4b\u503c\u3001\u6bd4\u8f83\u3001\u8f6c\u79fb&#xff09;\u90fd\u6709\u786e\u5207\u5b9a\u4e49&#xff1b;<\/li>\n<li>\u8f93\u5165 \u2705&#xff1a;\u96f6\u4e2a\u8f93\u5165&#xff08;\u95ee\u9898\u53c2\u6570\u5185\u5d4c\u5728\u7b97\u6cd5\u4e2d&#xff09;&#xff1b;<\/li>\n<li>\u8f93\u51fa \u2705&#xff1a;\u8f93\u51fa\u6bcf\u7ec4\u89e3\u53ca\u89e3\u7684\u603b\u6570 t&#xff1b;<\/li>\n<li>\u53ef\u884c\u6027 \u2705&#xff1a;\u52a0\u6cd5\u3001\u4e58\u6cd5\u3001\u6bd4\u8f83\u90fd\u662f\u57fa\u672c\u53ef\u6267\u884c\u64cd\u4f5c\u3002<\/li>\n<\/ul>\n<p>\u6240\u4ee5\u8fd9\u662f\u4e00\u4e2a\u5408\u683c\u7684\u7b97\u6cd5\u3002\u5b83\u867d\u7136&#034;\u7b28&#034;&#xff08;\u7a77\u4e3e&#xff09;&#xff0c;\u4f46\u7b80\u5355\u3001\u6e05\u6670\u3001\u5fc5\u7136\u6b63\u786e\u2014\u2014\u5728\u95ee\u9898\u89c4\u6a21\u5c0f\u7684\u65f6\u5019&#xff0c;\u7a77\u4e3e\u6cd5\u5f80\u5f80\u662f\u6700\u7a33\u59a5\u7684\u9009\u62e9\u3002\u8fd9\u4e5f\u4f53\u73b0\u4e86\u7b97\u6cd5\u8bbe\u8ba1\u4e2d&#034;\u5148\u4fdd\u8bc1\u6b63\u786e&#xff0c;\u518d\u8ffd\u6c42\u9ad8\u6548&#034;\u7684\u4e00\u822c\u539f\u5219\u3002<\/p>\n<h4>6.6 \u80fd\u5426\u6709\u5176\u4ed6\u63cf\u8ff0\u65b9\u6cd5&#xff1f;<\/h4>\n<p>\u5f53\u7136\u53ef\u4ee5&#xff01;\u7b97\u6cd5\u662f\u5bf9\u89e3\u9898\u8fc7\u7a0b\u7684\u7cbe\u786e\u63cf\u8ff0&#xff0c;\u4e14\u9700\u8981\u4f7f\u7528\u67d0\u79cd\u65b9\u6cd5\u5c06\u5176\u8868\u793a\u51fa\u6765\u3002\u63cf\u8ff0\u7b97\u6cd5\u7684\u5e38\u7528\u65b9\u6cd5\u6709\u4e09\u79cd&#xff1a;<\/p>\n<li>\u81ea\u7136\u8bed\u8a00&#xff1a;\u5982\u4e0a\u9762\u7684\u2460~\u246b\u6b65\u9aa4&#xff0c;\u4f18\u70b9\u662f\u901a\u4fd7\u6613\u61c2&#xff0c;\u7f3a\u70b9\u662f\u5197\u957f\u3001\u5bb9\u6613\u4ea7\u751f\u6b67\u4e49\u3001\u4e0d\u65b9\u4fbf\u8868\u793a\u5206\u652f\u5d4c\u5957&#xff1b;<\/li>\n<li>\u6d41\u7a0b\u56fe&#xff1a;\u7528\u56fe\u5f62\u7b26\u53f7&#xff08;\u8d77\u6b62\u6846\u3001\u5904\u7406\u6846\u3001\u5224\u65ad\u6846\u3001\u6d41\u7a0b\u7ebf&#xff09;\u76f4\u89c2\u8868\u793a\u6267\u884c\u6d41\u7a0b&#xff0c;\u4f18\u70b9\u662f\u5f62\u8c61\u76f4\u89c2&#xff0c;\u7f3a\u70b9\u662f\u7ed8\u5236\u9ebb\u70e6\u3001\u4fee\u6539\u4e0d\u4fbf\u3001\u5927\u578b\u7b97\u6cd5\u7684\u6d41\u7a0b\u56fe\u4f1a\u975e\u5e38\u590d\u6742&#xff1b;<\/li>\n<li>\u4f2a\u4ee3\u7801&#xff1a;\u4ecb\u4e8e\u81ea\u7136\u8bed\u8a00\u548c\u7a0b\u5e8f\u8bbe\u8ba1\u8bed\u8a00\u4e4b\u95f4&#xff0c;\u7528\u7c7b\u4f3c\u7f16\u7a0b\u8bed\u8a00\u7684\u8bed\u6cd5\u4f46\u4e0d\u62d8\u6ce5\u4e8e\u7ec6\u8282&#xff0c;\u4f18\u70b9\u662f\u7ed3\u6784\u6e05\u6670\u3001\u7b80\u6d01\u3001\u4fbf\u4e8e\u8f6c\u5316\u4e3a\u771f\u5b9e\u4ee3\u7801&#xff0c;\u662f\u6559\u6750\u548c\u8bba\u6587\u4e2d\u6700\u5e38\u7528\u7684\u7b97\u6cd5\u63cf\u8ff0\u65b9\u5f0f\u3002<\/li>\n<p>\u540c\u4e00\u4e2a\u7a77\u4e3e\u7b97\u6cd5\u7528\u4f2a\u4ee3\u7801\u53ef\u4ee5\u5199\u6210&#xff1a;<\/p>\n<p>Algorithm Enumerate()<br \/>\n    t \u2190 0<br \/>\n    for x \u2190 1 to 6 do<br \/>\n        for y \u2190 1 to 8 do<br \/>\n            for z \u2190 1 to 9 do<br \/>\n                if 6x &#043; 5y &#043; 4z &#061; 50 then<br \/>\n                    t \u2190 t &#043; 1<br \/>\n                    output(x, y, z)<br \/>\n    output(t)<\/p>\n<p>\u5bf9\u6bd4\u4e4b\u4e0b\u4e0d\u96be\u53d1\u73b0&#xff1a;\u4f2a\u4ee3\u7801\u628a12\u4e2a\u81ea\u7136\u8bed\u8a00\u6b65\u9aa4\u538b\u7f29\u6210\u4e86\u4e0d\u523010\u884c&#xff0c;\u4e14\u5faa\u73af\u5d4c\u5957\u7ed3\u6784\u4e00\u76ee\u4e86\u7136\u3002\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u8981\u5b66\u4f2a\u4ee3\u7801\u2014\u2014\u5b83\u662f\u7b97\u6cd5\u4e16\u754c\u7684&#034;\u901a\u7528\u8bed&#034;\u3002<\/p>\n<hr \/>\n<h3>\u4e03\u3001\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5&#xff08;\u8f97\u8f6c\u76f8\u9664\u6cd5&#xff09;\u4e0e\u7279\u5f81\u8fa8\u6790<\/h3>\n<h4>7.1 \u7b97\u6cd5\u63cf\u8ff0<\/h4>\n<p>\u5728\u300a\u51e0\u4f55\u539f\u672c\u300b\u4e00\u4e66\u4e2d&#xff0c;\u6b27\u51e0\u91cc\u5f97&#xff08;Euclid&#xff09; \u9610\u8ff0\u4e86\u6c42\u4e24\u4e2a\u6b63\u6574\u6570\u6700\u5927\u516c\u7ea6\u6570&#xff08;GCD&#xff09;\u7684\u8fc7\u7a0b&#xff0c;\u8fd9\u5c31\u662f\u8457\u540d\u7684\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u2014\u2014\u8f97\u8f6c\u76f8\u9664\u6cd5\u3002\u5b83\u662f\u4eba\u7c7b\u5386\u53f2\u4e0a\u6700\u53e4\u8001\u7684\u7b97\u6cd5\u4e4b\u4e00&#xff08;\u7ea6\u516c\u5143\u524d300\u5e74&#xff09;&#xff0c;\u81f3\u4eca\u4ecd\u662f\u6559\u79d1\u4e66\u7ea7\u7684\u7ecf\u5178\u3002<\/p>\n<p>\u8bbe\u7ed9\u5b9a\u7684\u4e24\u4e2a\u6b63\u6574\u6570\u4e3a m \u548c n&#xff0c;\u6c42\u5b83\u4eec\u7684\u6700\u5927\u516c\u7ea6\u6570\u7684\u6b65\u9aa4\u5982\u4e0b&#xff1a;<\/p>\n<p>1\u3001 \u4ee5 m \u9664\u4ee5 n&#xff0c;\u4ee4\u6240\u5f97\u7684\u4f59\u6570\u4e3a R\u3002 2\u3001 \u82e5 R&#061;0&#xff0c;\u5219\u8f93\u51fa\u7ed3\u679c n&#xff0c;\u7b97\u6cd5\u7ed3\u675f&#xff1b;\u5426\u5219&#xff0c;\u7ee7\u7eed\u6b65\u9aa43\u3002 3\u3001 \u4ee4 m&#061;n&#xff0c;n&#061;R&#xff0c;\u5e76\u8fd4\u56de\u6b65\u9aa41\u7ee7\u7eed\u8fdb\u884c\u3002<\/p>\n<p>\u7528\u4f2a\u4ee3\u7801\u8868\u793a&#xff1a;<\/p>\n<p>Algorithm gcd(m, n)<br \/>\n    while n \u2260 0 do<br \/>\n        R \u2190 m mod n<br \/>\n        m \u2190 n<br \/>\n        n \u2190 R<br \/>\n    return m<\/p>\n<p>\u5176\u6570\u5b66\u539f\u7406\u662f&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        gcd<\/p>\n<p>        \u2061<\/p>\n<p>        (<\/p>\n<p>        m<\/p>\n<p>        ,<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        gcd<\/p>\n<p>        \u2061<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        ,<\/p>\n<p>        m<br \/>\n       \u200a<\/p>\n<p>          m<\/p>\n<p>          o<\/p>\n<p>          d<\/p>\n<p>       \u200a<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       \\\\gcd(m, n) &#061; \\\\gcd(n, m \\\\bmod n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\"><span style=\"margin-right: 0.0139em\">g<\/span>cd<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\"><span style=\"margin-right: 0.0139em\">g<\/span>cd<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.0556em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\"><span class=\"mord\"><span class=\"mord mathrm\">mod<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.0556em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u4e0d\u65ad\u7528&#034;\u9664\u6570\u3001\u4f59\u6570&#034;\u66ff\u6362&#034;\u88ab\u9664\u6570\u3001\u9664\u6570&#034;&#xff0c;\u4f59\u6570\u4e25\u683c\u9012\u51cf\u4e14\u975e\u8d1f&#xff0c;\u6240\u4ee5\u5fc5\u7136\u5728\u6709\u9650\u6b65\u5185\u53d8\u62100&#xff0c;\u6b64\u65f6\u7684\u9664\u6570\u5c31\u662f\u6700\u5927\u516c\u7ea6\u6570\u3002<\/p>\n<h4>7.2 \u7279\u5f81\u8fa8\u6790\u586b\u7a7a\u9898&#xff08;\u8003\u8bd5\u539f\u9898\u7ea7&#xff09;<\/h4>\n<p>\u9898\u76ee1&#xff1a;\u5728\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u4e2d&#xff0c;\u6b65\u9aa41\u660e\u786e\u89c4\u5b9a&#034;\u4ee5 m \u9664\u4ee5 n&#xff0c;\u4ee4\u6240\u5f97\u7684\u4f59\u6570\u4e3a R&#034;&#xff0c;\u8fd9\u4f53\u73b0\u4e86\u7b97\u6cd5\u7684 ____ \u7279\u5f81\u3002<\/p>\n<p>\u7b54\u6848&#xff1a;\u786e\u5b9a\u6027\u3002 \u89e3\u6790&#xff1a;\u6b65\u9aa41\u5bf9&#034;\u505a\u4ec0\u4e48&#034;&#xff08;m\u9664\u4ee5n&#xff09;\u3001\u201c\u7ed3\u679c\u53eb\u4ec0\u4e48\u201d&#xff08;\u4f59\u6570\u8bb0\u4e3aR&#xff09;\u90fd\u7ed9\u51fa\u4e86\u786e\u5207\u7684\u5b9a\u4e49&#xff0c;\u6ca1\u6709\u4efb\u4f55\u6a21\u68f1\u4e24\u53ef\u7684\u5730\u65b9\u2014\u2014\u8c01\u9664\u4ee5\u8c01\u3001\u4f59\u6570\u5982\u4f55\u547d\u540d\u90fd\u89c4\u5b9a\u5f97\u6e05\u6e05\u695a\u695a&#xff0c;\u8fd9\u6b63\u662f\u786e\u5b9a\u6027\u7684\u4f53\u73b0\u3002<\/p>\n<p>\u9898\u76ee2&#xff1a;\u5728\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u4e2d&#xff0c;\u6b65\u9aa42&#034;\u82e5 R&#061;0&#xff0c;\u5219\u8f93\u51fa\u7ed3\u679c n&#xff0c;\u7b97\u6cd5\u7ed3\u675f&#xff1b;\u5426\u5219&#xff0c;\u7ee7\u7eed\u6b65\u9aa43&#034;&#xff0c;\u8868\u660e\u8fd9\u4e2a\u7b97\u6cd5\u6709\u7ed3\u679c\u8f93\u51fa&#xff0c;\u5e76\u4e14\u6709\u7ed3\u675f\u6761\u4ef6&#xff0c;\u8fd9\u4f53\u73b0\u4e86\u7b97\u6cd5\u7684 ____ \u548c ____ \u7279\u5f81\u3002<\/p>\n<p>\u7b54\u6848&#xff1a;\u6570\u636e\u8f93\u51fa\u3001\u6709\u7a77\u6027\u3002 \u89e3\u6790&#xff1a;&#034;\u8f93\u51fa\u7ed3\u679c n&#034;\u5bf9\u5e94\u6570\u636e\u8f93\u51fa\u7279\u5f81&#xff08;\u7b97\u6cd5\u6709\u81f3\u5c11\u4e00\u4e2a\u8f93\u51fa&#xff09;&#xff1b;&#034;\u7b97\u6cd5\u7ed3\u675f&#034;\u7ed9\u51fa\u4e86\u660e\u786e\u7684\u7ec8\u6b62\u6761\u4ef6&#xff0c;\u4f59\u6570\u5e8f\u5217\u4e25\u683c\u9012\u51cf\u4fdd\u8bc1\u4e86\u4e00\u5b9a\u80fd\u5230\u8fbe R&#061;0 \u4ece\u800c\u505c\u673a&#xff0c;\u5bf9\u5e94\u6709\u7a77\u6027\u7279\u5f81\u3002<\/p>\n<p>\u9898\u76ee3&#xff1a;\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u7684\u4e09\u4e2a\u6b65\u9aa4\u90fd\u662f\u53ef\u4ee5\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u7684&#xff0c;\u90fd\u662f\u53ef\u6267\u884c\u7684\u64cd\u4f5c\u6b65\u9aa4&#xff0c;\u8fd9\u4f53\u73b0\u4e86\u7b97\u6cd5\u7684 ____ \u7279\u5f81\u3002<\/p>\n<p>\u7b54\u6848&#xff1a;\u53ef\u884c\u6027\u3002 \u89e3\u6790&#xff1a;\u6c42\u4f59\u3001\u8d4b\u503c\u3001\u6bd4\u8f83\u3001\u8df3\u8f6c&#xff0c;\u6bcf\u4e00\u6b65\u90fd\u662f\u53ef\u4ee5\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u7684\u57fa\u672c\u64cd\u4f5c&#xff0c;\u8fd9\u6b63\u662f\u53ef\u884c\u6027\u7684\u5b9a\u4e49\u3002<\/p>\n<h4>7.3 \u7b54\u9898\u6280\u5de7\u603b\u7ed3<\/h4>\n<p>\u8fd9\u7c7b&#034;\u7ed9\u4e00\u6bb5\u7b97\u6cd5\u63cf\u8ff0&#xff0c;\u95ee\u4f53\u73b0\u4e86\u54ea\u4e9b\u7279\u5f81&#034;\u7684\u9898\u76ee&#xff0c;\u6293\u5173\u952e\u8bcd\u5373\u53ef&#xff1a;<\/p>\n<table>\n<tr>\u63cf\u8ff0\u4e2d\u7684\u5173\u952e\u8bcd\u5bf9\u5e94\u7279\u5f81<\/tr>\n<tbody>\n<tr>\n<td>\u201c\u660e\u786e\u89c4\u5b9a\u201d\u201c\u786e\u5207\u5b9a\u4e49\u201d\u201c\u4e0d\u542b\u7cca\u201d<\/td>\n<td>\u786e\u5b9a\u6027<\/td>\n<\/tr>\n<tr>\n<td>\u201c\u7b97\u6cd5\u7ed3\u675f\u201d\u201c\u7ec8\u6b62\u6761\u4ef6\u201d\u201c\u6709\u9650\u6b65\u201d<\/td>\n<td>\u6709\u7a77\u6027<\/td>\n<\/tr>\n<tr>\n<td>\u201c\u8f93\u51fa\u7ed3\u679c\u201d\u201c\u6253\u5370\u201d\u201c\u8fd4\u56de\u201d<\/td>\n<td>\u6570\u636e\u8f93\u51fa<\/td>\n<\/tr>\n<tr>\n<td>\u201c\u8f93\u5165m\u548cn\u201d\u201c\u7ed9\u5b9a\u6570\u636e\u201d<\/td>\n<td>\u6570\u636e\u8f93\u5165<\/td>\n<\/tr>\n<tr>\n<td>\u201c\u53ef\u4ee5\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u201d\u201c\u53ef\u6267\u884c\u7684\u64cd\u4f5c\u201d<\/td>\n<td>\u53ef\u884c\u6027<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u516b\u3001\u3010\u4f8b1-2\u3011\u8fdd\u53cd\u7b97\u6cd5\u7279\u5f81\u7684\u4e24\u6bb5\u4ee3\u7801<\/h3>\n<h4>8.1 \u9898\u76ee<\/h4>\n<p>\u6709\u4e0b\u5217\u4e24\u6bb5\u63cf\u8ff0&#xff0c;\u5b83\u4eec\u5747\u4e0d\u80fd\u6ee1\u8db3\u7b97\u6cd5\u7684\u7279\u5f81&#xff0c;\u8bd5\u95ee\u5b83\u4eec\u8fdd\u53cd\u4e86\u7b97\u6cd5\u7684\u54ea\u4e9b\u7279\u5f81&#xff1f;<\/p>\n<p>\u63cf\u8ff01&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">exam1<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">;<\/span><br \/>\n    n <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">%<\/span> <span class=\"token number\">2<\/span> <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        n <span class=\"token operator\">&#061;<\/span> n <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;%d\\\\n&#034;<\/span><span class=\"token punctuation\">,<\/span> n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u63cf\u8ff02&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">exam2<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> x<span class=\"token punctuation\">,<\/span> y<span class=\"token punctuation\">;<\/span><br \/>\n    y <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    x <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">5<\/span> <span class=\"token operator\">\/<\/span> y<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;%d,%d\\\\n&#034;<\/span><span class=\"token punctuation\">,<\/span> x<span class=\"token punctuation\">,<\/span> y<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>8.2 \u8003\u70b9\u89e3\u6790<\/h4>\n<p>\u63cf\u8ff01&#xff1a;\u8fdd\u53cd\u6709\u7a77\u6027 &#x1f50d;<\/p>\n<p>\u8ffd\u8e2a\u4e00\u4e0b\u6267\u884c\u8fc7\u7a0b&#xff1a;<\/p>\n<ul>\n<li>\u521d\u59cb n &#061; 2&#xff0c;2 % 2 &#061;&#061; 0 \u6210\u7acb&#xff0c;\u8fdb\u5165\u5faa\u73af&#xff1b;<\/li>\n<li>n &#061; n &#043; 2 &#061; 4&#xff0c;4 % 2 &#061;&#061; 0 \u4ecd\u7136\u6210\u7acb&#xff0c;\u7ee7\u7eed\u5faa\u73af&#xff1b;<\/li>\n<li>n &#061; 6, 8, 10, \u2026&#xff0c;n \u6c38\u8fdc\u662f\u5076\u6570&#xff0c;\u5faa\u73af\u6761\u4ef6\u6c38\u8fdc\u6210\u7acb&#xff1b;<\/li>\n<li>printf \u90a3\u4e00\u884c\u6c38\u8fdc\u6267\u884c\u4e0d\u5230&#xff0c;\u7b97\u6cd5\u65e0\u6cd5\u7ec8\u6b62\u3002<\/li>\n<\/ul>\n<p>\u4e00\u4e2a\u6c38\u8fdc\u505c\u4e0d\u4e0b\u6765\u7684&#034;\u7b97\u6cd5&#034;\u8fdd\u53cd\u4e86\u6709\u7a77\u6027&#xff08;\u6709\u9650\u6027&#xff09;\u7279\u5f81\u3002<\/p>\n<p>\u63cf\u8ff02&#xff1a;\u8fdd\u53cd\u53ef\u884c\u6027 &#x1f50d;<\/p>\n<ul>\n<li>y &#061; 0&#xff0c;\u63a5\u7740\u6267\u884c x &#061; 5 \/ y&#xff0c;\u5373 \u9664\u4ee5\u96f6&#xff1b;<\/li>\n<li>\u9664\u4ee5\u96f6\u5728\u6570\u5b66\u4e0a\u6ca1\u6709\u5b9a\u4e49&#xff0c;\u5728\u8ba1\u7b97\u673a\u4e0a\u662f\u975e\u6cd5\u64cd\u4f5c&#xff08;\u4f1a\u89e6\u53d1\u8fd0\u884c\u65f6\u9519\u8bef\/\u5f02\u5e38&#xff0c;\u7a0b\u5e8f\u5d29\u6e83&#xff09;&#xff1b;<\/li>\n<li>\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u8fd9\u4e2a\u6b65\u9aa4\u4e0d\u662f\u53ef\u4ee5\u5728\u6709\u9650\u65f6\u95f4\u5185\u5b8c\u6210\u7684\u53ef\u6267\u884c\u64cd\u4f5c\u3002<\/li>\n<\/ul>\n<p>\u56e0\u6b64\u63cf\u8ff02\u8fdd\u53cd\u4e86\u53ef\u884c\u6027\u7279\u5f81\u3002<\/p>\n<h4>8.3 \u6807\u51c6\u89e3\u7b54<\/h4>\n<p>\u89e3&#xff1a;&#xff08;1&#xff09;\u63cf\u8ff01\u662f\u4e00\u4e2a\u6b7b\u5faa\u73af&#xff08;n \u521d\u59cb\u4e3a\u5076\u65702&#xff0c;\u6bcf\u6b21\u52a02\u4ecd\u4e3a\u5076\u6570&#xff0c;\u5faa\u73af\u6761\u4ef6 n%2&#061;&#061;0 \u6052\u6210\u7acb&#xff09;&#xff0c;\u6c38\u8fdc\u4e0d\u4f1a\u6267\u884c\u5faa\u73af\u4f53\u4e4b\u540e\u7684\u8f93\u51fa\u8bed\u53e5&#xff0c;\u8fdd\u53cd\u4e86\u7b97\u6cd5\u7684\u6709\u7a77\u6027\u7279\u5f81\u3002<\/p>\n<p>&#xff08;2&#xff09;\u63cf\u8ff02\u4e2d y&#061;0&#xff0c;\u6267\u884c x&#061;5\/y \u65f6\u51fa\u73b0\u9664\u96f6\u9519\u8bef&#xff0c;\u8be5\u64cd\u4f5c\u4e0d\u53ef\u6267\u884c&#xff0c;\u8fdd\u53cd\u4e86\u7b97\u6cd5\u7684\u53ef\u884c\u6027\u7279\u5f81\u3002<\/p>\n<h4>8.4 \u5bf9\u6bd4\u8bb0\u5fc6<\/h4>\n<table>\n<tr>\u63cf\u8ff01\u63cf\u8ff02<\/tr>\n<tbody>\n<tr>\n<td>\u8868\u9762\u73b0\u8c61<\/td>\n<td>\u6b7b\u5faa\u73af<\/td>\n<td>\u9664\u96f6\u5d29\u6e83<\/td>\n<\/tr>\n<tr>\n<td>\u8fdd\u53cd\u7279\u5f81<\/td>\n<td>\u6709\u7a77\u6027<\/td>\n<td>\u53ef\u884c\u6027<\/td>\n<\/tr>\n<tr>\n<td>\u672c\u8d28\u533a\u522b<\/td>\n<td>\u6bcf\u4e00\u6b65\u90fd\u80fd\u6267\u884c&#xff0c;\u4f46\u6574\u4f53\u505c\u4e0d\u4e0b\u6765<\/td>\n<td>\u67d0\u4e00\u6b65\u6839\u672c\u65e0\u6cd5\u6267\u884c<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u518d\u6b21\u5f3a\u8c03\u524d\u9762\u63d0\u5230\u7684\u8fa8\u6790\u8981\u70b9&#xff1a;\u6709\u7a77\u6027\u770b\u6574\u4f53&#xff08;\u80fd\u5426\u7ec8\u6b62&#xff09;&#xff0c;\u53ef\u884c\u6027\u770b\u5355\u6b65&#xff08;\u80fd\u5426\u6267\u884c&#xff09;\u3002\u8fd9\u4e24\u6bb5\u4ee3\u7801\u6070\u597d\u662f\u4e00\u5bf9\u7edd\u4f73\u7684\u5bf9\u6bd4\u6837\u4f8b&#xff0c;\u8003\u8bd5\u7ecf\u5e38\u6210\u5bf9\u51fa\u73b0&#xff0c;\u52a1\u5fc5\u8bb0\u7262\u3002<\/p>\n<hr \/>\n<h3>\u4e5d\u3001\u63cf\u8ff0\u7b97\u6cd5\u7684\u4e09\u79cd\u5e38\u7528\u65b9\u6cd5&#xff08;\u5c55\u5f00&#xff09;<\/h3>\n<p>\u524d\u9762\u5df2\u7ecf\u63d0\u5230&#xff0c;\u63cf\u8ff0\u7b97\u6cd5\u7684\u5e38\u7528\u65b9\u6cd5\u6709\u4e09\u79cd&#xff0c;\u8fd9\u91cc\u518d\u7cfb\u7edf\u5c55\u5f00\u4e00\u4e0b&#xff0c;\u65b9\u4fbf\u5927\u5bb6\u505a\u7b80\u7b54\u9898&#xff1a;<\/p>\n<h4>9.1 \u81ea\u7136\u8bed\u8a00<\/h4>\n<ul>\n<li>\u4f18\u70b9&#xff1a;\u901a\u4fd7\u6613\u61c2&#xff0c;\u4e0d\u9700\u8981\u4efb\u4f55\u7f16\u7a0b\u57fa\u7840&#xff1b;<\/li>\n<li>\u7f3a\u70b9&#xff1a;\u6587\u5b57\u5197\u957f&#xff0c;\u5bb9\u6613\u4ea7\u751f\u6b67\u4e49&#xff08;\u8fdd\u53cd\u786e\u5b9a\u6027\u7684\u98ce\u9669&#xff09;&#xff0c;\u5206\u652f\u548c\u5faa\u73af\u5d4c\u5957\u5c42\u6b21\u591a\u65f6\u63cf\u8ff0\u56f0\u96be&#xff1b;<\/li>\n<li>\u9002\u7528\u573a\u666f&#xff1a;\u5411\u975e\u4e13\u4e1a\u4eba\u5458\u89e3\u91ca\u7b97\u6cd5\u601d\u60f3&#xff0c;\u6216\u50cf&#034;6x&#043;5y&#043;4z&#061;50&#034;\u8fd9\u79cd\u6b65\u9aa4\u7b80\u5355\u7684\u7b97\u6cd5\u3002<\/li>\n<\/ul>\n<h4>9.2 \u6d41\u7a0b\u56fe<\/h4>\n<ul>\n<li>\u57fa\u672c\u7b26\u53f7&#xff1a;\n<ul>\n<li>\u5706\u89d2\u77e9\u5f62&#xff08;\u692d\u5706&#xff09;&#xff1a;\u5f00\u59cb\/\u7ed3\u675f\u6846&#xff1b;<\/li>\n<li>\u77e9\u5f62&#xff1a;\u5904\u7406\u6846&#xff08;\u6267\u884c\u64cd\u4f5c&#xff09;&#xff1b;<\/li>\n<li>\u83f1\u5f62&#xff1a;\u5224\u65ad\u6846&#xff08;\u6761\u4ef6\u5206\u652f&#xff0c;\u4e24\u4e2a\u51fa\u53e3&#034;\u662f\/\u5426&#034;&#xff09;&#xff1b;<\/li>\n<li>\u7bad\u5934&#xff08;\u6d41\u7a0b\u7ebf&#xff09;&#xff1a;\u8868\u793a\u6267\u884c\u987a\u5e8f&#xff1b;<\/li>\n<li>\u5e73\u884c\u56db\u8fb9\u5f62&#xff1a;\u8f93\u5165\/\u8f93\u51fa\u6846\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\u4f18\u70b9&#xff1a;\u5f62\u8c61\u76f4\u89c2&#xff0c;\u6d41\u7a0b\u4e00\u76ee\u4e86\u7136&#xff1b;<\/li>\n<li>\u7f3a\u70b9&#xff1a;\u7ed8\u5236\u548c\u4fee\u6539\u9ebb\u70e6&#xff0c;\u590d\u6742\u7b97\u6cd5\u7684\u6d41\u7a0b\u56fe\u5e9e\u5927\u96be\u8bfb&#xff0c;\u4e14\u6d41\u7a0b\u56fe\u662f&#034;\u56fe\u5f62&#034;\u800c\u975e&#034;\u6587\u672c&#034;&#xff0c;\u4e0d\u4fbf\u4e8e\u7248\u672c\u7ba1\u7406\u548c\u4f20\u64ad&#xff1b;<\/li>\n<li>\u9002\u7528\u573a\u666f&#xff1a;\u6559\u5b66\u6f14\u793a\u3001\u7b80\u5355\u6d41\u7a0b\u7684\u4e1a\u52a1\u8bf4\u660e\u3002<\/li>\n<\/ul>\n<h4>9.3 \u4f2a\u4ee3\u7801<\/h4>\n<ul>\n<li>\u7279\u70b9&#xff1a;\u501f\u7528\u7a0b\u5e8f\u8bbe\u8ba1\u8bed\u8a00\u7684\u9aa8\u67b6&#xff08;if\/else\u3001for\/while\u3001\u8d4b\u503c\u3001\u51fd\u6570\u8c03\u7528&#xff09;&#xff0c;\u4f46\u4e0d\u53d7\u5177\u4f53\u8bed\u8a00\u8bed\u6cd5\u7ea6\u675f&#xff0c;\u53ef\u4ee5\u7528\u6570\u5b66\u7b26\u53f7\u548c\u81ea\u7136\u8bed\u8a00\u6df7\u6392&#xff1b;<\/li>\n<li>\u4f18\u70b9&#xff1a;\u7ed3\u6784\u6e05\u6670\u3001\u7b80\u6d01\u7d27\u51d1\u3001\u7a81\u51fa\u7b97\u6cd5\u903b\u8f91\u3001\u5c4f\u853d\u8bed\u8a00\u7ec6\u8282\u3001\u6613\u4e8e\u7ffb\u8bd1\u6210\u4efb\u4f55\u4e00\u95e8\u7f16\u7a0b\u8bed\u8a00&#xff1b;<\/li>\n<li>\u7f3a\u70b9&#xff1a;\u6ca1\u6709\u7edf\u4e00\u6807\u51c6&#xff0c;\u4e0d\u540c\u6559\u6750\u5199\u6cd5\u7565\u6709\u5dee\u5f02&#xff1b;\u5bf9\u5b8c\u5168\u4e0d\u61c2\u7f16\u7a0b\u7684\u4eba\u4e0d\u53cb\u597d&#xff1b;<\/li>\n<li>\u9002\u7528\u573a\u666f&#xff1a;\u6559\u6750\u3001\u5b66\u672f\u8bba\u6587\u3001\u7b97\u6cd5\u7ade\u8d5b\u9898\u89e3\u2014\u2014\u8fd9\u662f\u672c\u8bfe\u7a0b\u540e\u7eed\u7ae0\u8282\u7684\u4e3b\u8981\u63cf\u8ff0\u5de5\u5177&#xff0c;\u5efa\u8bae\u4ece\u73b0\u5728\u5f00\u59cb\u5c31\u7ec3\u4e60\u4e66\u5199\u89c4\u8303\u7684\u4f2a\u4ee3\u7801\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u5341\u3001\u7528C\/C&#043;&#043;\u63cf\u8ff0\u7b97\u6cd5\u7684\u4e00\u822c\u5f62\u5f0f<\/h3>\n<h4>10.1 \u6c42 1&#043;2&#043;\u2026&#043;n \u7684\u7b97\u6cd5<\/h4>\n<p>\u4ee5\u8bbe\u8ba1\u6c42 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          1<\/p>\n<p>          &#043;<\/p>\n<p>          2<\/p>\n<p>          &#043;<\/p>\n<p>          \u22ef<\/p>\n<p>          &#043;<\/p>\n<p>          n<\/p>\n<p>       \\\\boldsymbol{1&#043;2&#043;\\\\cdots&#043;n}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7778em;vertical-align: -0.1333em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbf\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathbf\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin mathbf\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord boldsymbol\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u503c\u7684\u7b97\u6cd5\u4e3a\u4f8b&#xff0c;\u8bf4\u660e C\/C&#043;&#043; \u8bed\u8a00\u63cf\u8ff0\u7b97\u6cd5\u7684\u4e00\u822c\u5f62\u5f0f\u3002\u5176\u6838\u5fc3\u7ea6\u5b9a\u662f&#xff1a;<\/p>\n<ul>\n<li>\u7528\u51fd\u6570\u7684\u8fd4\u56de\u503c\u8868\u793a\u7b97\u6cd5\u80fd\u5426\u6b63\u786e\u6267\u884c&#xff08;\u5982 bool&#xff1a;true \u8868\u793a\u6210\u529f&#xff0c;false \u8868\u793a\u5931\u8d25&#xff09;&#xff1b;<\/li>\n<li>\u7528\u5f62\u53c2\u8868\u793a\u7b97\u6cd5\u7684\u8f93\u5165\u8f93\u51fa&#xff08;\u8f93\u5165\u578b\u5f62\u53c2\u4f20\u6570\u636e\u8fdb\u6765&#xff0c;\u8f93\u51fa\u578b\u5f62\u53c2\u628a\u7ed3\u679c\u5e26\u51fa\u53bb&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u6559\u6750\u7ed9\u51fa\u7684\u7b97\u6cd5\u6846\u67b6\u5982\u4e0b&#xff1a;<\/p>\n<p>bool <span class=\"token function\">fun<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> s<span class=\"token punctuation\">)<\/span>          <span class=\"token comment\">\/\/ \u8fd4\u56de\u503c&#xff1a;\u7b97\u6cd5\u662f\u5426\u6b63\u786e\u6267\u884c&#xff1b;\u5f62\u53c2&#xff1a;n\u4e3a\u8f93\u5165&#xff0c;s\u4e3a\u8f93\u51fa<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">&lt;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">return<\/span> false<span class=\"token punctuation\">;<\/span>    <span class=\"token comment\">\/\/ \u975e\u6cd5\u8f93\u5165&#xff0c;\u7b97\u6cd5\u6267\u884c\u5931\u8d25<\/span><br \/>\n    s <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        s <span class=\"token operator\">&#043;&#061;<\/span> i<span class=\"token punctuation\">;<\/span>                 <span class=\"token comment\">\/\/ \u7d2f\u52a0\u6c42\u548c<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> true<span class=\"token punctuation\">;<\/span>                <span class=\"token comment\">\/\/ \u6b63\u5e38\u5b8c\u6210&#xff0c;\u8fd4\u56detrue<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">main<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> a <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">10<\/span><span class=\"token punctuation\">,<\/span> b <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token function\">fun<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;%d\\\\n&#034;<\/span><span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">else<\/span> <span class=\"token function\">printf<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u53c2\u6570\u9519\u8bef\\\\n&#034;<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u6807\u6ce8\u8bf4\u660e&#xff1a;<\/p>\n<ul>\n<li>bool \u8fd4\u56de\u503c \u2192 \u8868\u793a\u7b97\u6cd5\u6267\u884c\u72b6\u6001&#xff08;\u6210\u529f\/\u5931\u8d25&#xff09;&#xff1b;<\/li>\n<li>\u5f62\u53c2\u5217\u8868 (int n, int s) \u2192 n \u662f\u8f93\u5165\u53c2\u6570&#xff0c;s \u610f\u56fe\u4f5c\u4e3a\u8f93\u51fa\u53c2\u6570\u3002<\/li>\n<\/ul>\n<h4>10.2 \u26a0\ufe0f \u4e00\u4e2a\u503c\u5f97\u6df1\u6316\u7684\u7ec6\u8282&#xff1a;\u503c\u4f20\u9012\u7684\u9677\u9631<\/h4>\n<p>\u7ec6\u5fc3\u7684\u540c\u5b66\u53ef\u80fd\u5df2\u7ecf\u53d1\u73b0&#xff1a;\u4e0a\u9762 fun(int n, int s) \u4e2d&#xff0c;s \u662f\u503c\u4f20\u9012\u7684\u666e\u901a\u5f62\u53c2\u3002\u51fd\u6570\u5185\u90e8 s &#061; 0; s &#043;&#061; i; \u4fee\u6539\u7684\u53ea\u662f\u5b9e\u53c2 b \u7684\u4e00\u4efd\u526f\u672c&#xff0c;\u51fd\u6570\u8fd4\u56de\u540e main \u4e2d\u7684 b \u4f9d\u7136\u662f 0&#xff01;\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u8fd9\u6bb5\u4ee3\u7801\u867d\u7136\u5f62\u5f0f\u4e0a\u6f14\u793a\u4e86&#034;\u7528\u5f62\u53c2\u8868\u793a\u8f93\u5165\u8f93\u51fa&#034;&#xff0c;\u4f46\u8f93\u51fa\u5e76\u6ca1\u6709\u771f\u6b63\u5e26\u56de\u53bb\u3002<\/p>\n<p>\u8981\u60f3\u8ba9 s \u771f\u6b63\u4f5c\u4e3a\u8f93\u51fa\u53c2\u6570&#xff0c;\u5e94\u6539\u7528\u5f15\u7528\u4f20\u9012&#xff1a;<\/p>\n<p>bool <span class=\"token function\">fun<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> <span class=\"token operator\">&amp;<\/span>s<span class=\"token punctuation\">)<\/span>         <span class=\"token comment\">\/\/ s\u6539\u4e3a\u5f15\u7528\u7c7b\u578b<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">&lt;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">return<\/span> false<span class=\"token punctuation\">;<\/span><br \/>\n    s <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        s <span class=\"token operator\">&#043;&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> true<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u8fd9\u6837 main \u4e2d\u7684 b \u624d\u4f1a\u88ab\u8d4b\u503c\u4e3a 55&#xff08;\u5f53 a&#061;10 \u65f6&#xff09;\u3002<\/p>\n<p>&#x1f4a1; \u9884\u4e60\u542f\u793a&#xff1a;\u8fd9\u4e2a\u7ec6\u8282\u6070\u597d\u547c\u5e94\u4e86\u7b2c4\u8282\u4f8b1-1\u7684\u6559\u8bad\u2014\u2014&#034;\u5f62\u53c2\/\u8fd4\u56de\u503c\u7684\u8bbe\u8ba1&#034;\u76f4\u63a5\u51b3\u5b9a\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u3002\u6559\u6750\u5728\u540e\u7eed\u7ae0\u8282&#xff08;\u5982\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u4e2d\u7684 Solve(double a[][MAX], int m, int n, double &amp;s)&#xff09;\u5c31\u89c4\u8303\u5730\u4f7f\u7528\u4e86\u5f15\u7528\u53c2\u6570 double &amp;s\u3002\u5927\u5bb6\u5728\u5199\u4ee3\u7801\u65f6\u52a1\u5fc5\u5206\u6e05&#xff1a;\u503c\u4f20\u9012\u4f20\u526f\u672c&#xff0c;\u5f15\u7528\u4f20\u9012\u4f20\u672c\u4f53&#xff0c;\u6307\u9488\u4f20\u9012\u4f20\u5730\u5740\u3002<\/p>\n<h4>10.3 C\/C&#043;&#043;\u63cf\u8ff0\u7b97\u6cd5\u7684\u4e00\u822c\u5f62\u5f0f\u603b\u7ed3<\/h4>\n<p>\u4e00\u4e2a\u89c4\u8303\u7684\u7b97\u6cd5\u51fd\u6570\u5e94\u5f53\u5305\u542b&#xff1a;<\/p>\n<li>\u51fd\u6570\u540d&#xff1a;\u89c1\u540d\u77e5\u4e49&#xff08;\u5982 Find\u3001maxelem\u3001mergesort&#xff09;&#xff1b;<\/li>\n<li>\u5f62\u53c2\u5217\u8868&#xff1a;\u8f93\u5165\u53c2\u6570\u5728\u524d&#xff0c;\u8f93\u51fa\u53c2\u6570\u5728\u540e&#xff08;\u8f93\u51fa\u53c2\u6570\u7528\u5f15\u7528\u6216\u6307\u9488&#xff09;&#xff1b;<\/li>\n<li>\u8fd4\u56de\u503c&#xff1a;\u53ef\u4ee5\u643a\u5e26\u7ed3\u679c&#xff0c;\u4e5f\u53ef\u4ee5\u50cf bool \u4e00\u6837\u53ea\u8868\u793a\u6267\u884c\u72b6\u6001&#xff1b;<\/li>\n<li>\u51fd\u6570\u4f53&#xff1a;\u8fb9\u754c\/\u975e\u6cd5\u8f93\u5165\u68c0\u67e5 \u2192 \u6838\u5fc3\u8ba1\u7b97 \u2192 \u8fd4\u56de\u7ed3\u679c&#xff1b;<\/li>\n<li>\u5fc5\u8981\u7684\u6ce8\u91ca&#xff1a;\u8bf4\u660e\u7b97\u6cd5\u529f\u80fd\u3001\u53c2\u6570\u542b\u4e49\u3001\u590d\u6742\u5ea6\u7b49\u3002<\/li>\n<hr \/>\n<h3>\u5341\u4e00\u3001\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u8054\u7cfb\u4e0e\u533a\u522b<\/h3>\n<p>\u8fd9\u662f\u4e00\u9053\u7ecf\u5178\u7684\u7b80\u7b54\u9898&#xff0c;\u6559\u6750\u7684\u8868\u8ff0\u662f&#034;\u65e2\u6709\u8054\u7cfb\u53c8\u6709\u533a\u522b&#034;&#xff1a;<\/p>\n<h4>11.1 \u8054\u7cfb<\/h4>\n<p>\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u8bbe\u8ba1\u7684\u57fa\u7840\u3002 \u7b97\u6cd5\u7684\u64cd\u4f5c\u5bf9\u8c61\u662f\u6570\u636e\u7ed3\u6784&#xff0c;\u5728\u8bbe\u8ba1\u7b97\u6cd5\u65f6&#xff0c;\u8981\u6784\u5efa\u9002\u5408\u8fd9\u79cd\u7b97\u6cd5\u7684\u6570\u636e\u7ed3\u6784\u3002\u6570\u636e\u7ed3\u6784\u8bbe\u8ba1\u4e3b\u8981\u662f\u9009\u62e9\u6570\u636e\u7684\u5b58\u50a8\u65b9\u5f0f&#xff0c;\u5982\u786e\u5b9a\u6c42\u89e3\u95ee\u9898\u4e2d\u7684\u6570\u636e\u91c7\u7528\u6570\u7ec4\u5b58\u50a8\u8fd8\u662f\u91c7\u7528\u94fe\u8868\u5b58\u50a8\u7b49\u3002\u7b97\u6cd5\u8bbe\u8ba1\u5c31\u662f\u5728\u9009\u5b9a\u7684\u5b58\u50a8\u7ed3\u6784\u4e0a\u8bbe\u8ba1\u4e00\u4e2a\u6ee1\u8db3\u8981\u6c42\u7684\u597d\u7b97\u6cd5\u3002<\/p>\n<p>\u4e3e\u4e2a\u4f8b\u5b50\u4f53\u4f1a\u4e00\u4e0b&#xff1a;\u540c\u6837\u662f&#034;\u67e5\u627e&#034;\u64cd\u4f5c\u2014\u2014<\/p>\n<ul>\n<li>\u6570\u636e\u7528\u6709\u5e8f\u6570\u7ec4\u5b58\u50a8 \u2192 \u53ef\u4ee5\u8bbe\u8ba1\u4e8c\u5206\u67e5\u627e\u7b97\u6cd5&#xff0c;O(log n)&#xff1b;<\/li>\n<li>\u6570\u636e\u7528\u5355\u94fe\u8868\u5b58\u50a8 \u2192 \u53ea\u80fd\u987a\u5e8f\u67e5\u627e&#xff0c;O(n)&#xff0c;\u56e0\u4e3a\u94fe\u8868\u4e0d\u652f\u6301\u968f\u673a\u8bbf\u95ee&#xff1b;<\/li>\n<li>\u6570\u636e\u7528\u54c8\u5e0c\u8868\u5b58\u50a8 \u2192 \u5e73\u5747 O(1) \u67e5\u627e\u3002<\/li>\n<\/ul>\n<p>\u5b58\u50a8\u7ed3\u6784\u7684\u9009\u62e9\u76f4\u63a5\u51b3\u5b9a\u4e86\u4f60\u80fd\u8bbe\u8ba1\u51fa\u4ec0\u4e48\u6837\u7684\u7b97\u6cd5\u3001\u7b97\u6cd5\u80fd\u8fbe\u5230\u4ec0\u4e48\u6837\u7684\u6548\u7387\u3002\u8fd9\u5c31\u662f&#034;\u6570\u636e\u7ed3\u6784\u662f\u7b97\u6cd5\u8bbe\u8ba1\u7684\u57fa\u7840&#034;\u7684\u542b\u4e49\u3002\u8457\u540d\u79d1\u5b66\u5bb6\u6c83\u65af&#xff08;Niklaus Wirth&#xff09;\u7684\u540d\u8a00&#034;\u7a0b\u5e8f &#061; \u6570\u636e\u7ed3\u6784 &#043; \u7b97\u6cd5&#034;\u8bf4\u7684\u6b63\u662f\u8fd9\u79cd\u5bc6\u4e0d\u53ef\u5206\u7684\u5173\u7cfb\u3002<\/p>\n<h4>11.2 \u533a\u522b<\/h4>\n<p>\u6570\u636e\u7ed3\u6784\u5173\u6ce8\u7684\u662f\u6570\u636e\u7684\u903b\u8f91\u7ed3\u6784\u3001\u5b58\u50a8\u7ed3\u6784\u4ee5\u53ca\u57fa\u672c\u64cd\u4f5c&#xff1b;\u800c\u7b97\u6cd5\u66f4\u591a\u7684\u662f\u5173\u6ce8\u5982\u4f55\u5728\u6570\u636e\u7ed3\u6784\u7684\u57fa\u7840\u4e0a\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u3002\u7b97\u6cd5\u662f\u7f16\u7a0b\u601d\u60f3&#xff0c;\u6570\u636e\u7ed3\u6784\u5219\u662f\u8fd9\u4e9b\u601d\u60f3\u7684\u903b\u8f91\u57fa\u7840\u3002<\/p>\n<table>\n<tr>\u7ef4\u5ea6\u6570\u636e\u7ed3\u6784\u7b97\u6cd5<\/tr>\n<tbody>\n<tr>\n<td>\u7814\u7a76\u5bf9\u8c61<\/td>\n<td>\u6570\u636e\u672c\u8eab&#xff1a;\u903b\u8f91\u7ed3\u6784&#xff08;\u7ebf\u6027\/\u6811\u5f62\/\u56fe\u5f62&#xff09;\u3001\u5b58\u50a8\u7ed3\u6784&#xff08;\u987a\u5e8f\/\u94fe\u5f0f\/\u7d22\u5f15\/\u6563\u5217&#xff09;\u3001\u57fa\u672c\u8fd0\u7b97<\/td>\n<td>\u89e3\u51b3\u95ee\u9898\u7684\u6b65\u9aa4\u4e0e\u65b9\u6cd5&#xff1a;\u5982\u4f55\u52a0\u5de5\u6570\u636e\u5f97\u5230\u7b54\u6848<\/td>\n<\/tr>\n<tr>\n<td>\u56de\u7b54\u7684\u95ee\u9898<\/td>\n<td>\u201c\u6570\u636e\u600e\u4e48\u7ec4\u7ec7\u3001\u600e\u4e48\u5b58&#xff1f;\u201d<\/td>\n<td>\u201c\u95ee\u9898\u600e\u4e48\u89e3\u3001\u89e3\u5f97\u597d\u4e0d\u597d&#xff1f;\u201d<\/td>\n<\/tr>\n<tr>\n<td>\u89d2\u8272\u5b9a\u4f4d<\/td>\n<td>\u601d\u60f3\u7684\u903b\u8f91\u57fa\u7840&#xff08;\u9759\u6001\u9aa8\u67b6&#xff09;<\/td>\n<td>\u7f16\u7a0b\u601d\u60f3&#xff08;\u52a8\u6001\u7075\u9b42&#xff09;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>11.3 \u6c42\u89e3\u95ee\u9898\u7684\u5b8c\u6574\u6d41\u7a0b<\/h4>\n<p>\u6559\u6750\u7ed9\u51fa\u4e86\u7528\u8ba1\u7b97\u673a\u6c42\u89e3\u95ee\u9898\u7684\u4e00\u822c\u6d41\u7a0b\u56fe&#xff1a;<\/p>\n<p>\u5206\u6790\u6c42\u89e3\u95ee\u9898<br \/>\n    \u2193<br \/>\n\u9009\u62e9\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565<br \/>\n    \u2193<br \/>\n\u63cf\u8ff0\u7b97\u6cd5<br \/>\n    \u2193<br \/>\n\u8bc1\u660e\u7b97\u6cd5\u6b63\u786e\u6027<br \/>\n    \u2193<br \/>\n\u7b97\u6cd5\u5206\u6790<\/p>\n<p>\u8fd9\u4e94\u6b65\u662f\u4e00\u4e2a\u65b9\u6cd5\u8bba\u95ed\u73af&#xff1a;<\/p>\n<li>\u5206\u6790\u6c42\u89e3\u95ee\u9898&#xff1a;\u5f04\u6e05\u8f93\u5165\u662f\u4ec0\u4e48\u3001\u8f93\u51fa\u662f\u4ec0\u4e48\u3001\u7ea6\u675f\u6761\u4ef6\u6709\u54ea\u4e9b&#xff1b;<\/li>\n<li>\u9009\u62e9\u6570\u636e\u7ed3\u6784\u548c\u7b97\u6cd5\u8bbe\u8ba1\u7b56\u7565&#xff1a;\u51b3\u5b9a\u6570\u636e\u600e\u4e48\u5b58&#xff08;\u6570\u7ec4&#xff1f;\u94fe\u8868&#xff1f;\u6811&#xff1f;&#xff09;&#xff0c;\u7528\u4ec0\u4e48\u7b56\u7565\u89e3&#xff08;\u7a77\u4e3e&#xff1f;\u5206\u6cbb&#xff1f;\u8d2a\u5fc3&#xff1f;\u52a8\u6001\u89c4\u5212&#xff1f;&#xff09;\u2014\u2014\u540e\u9762\u7ae0\u8282\u5b66\u7684\u5404\u79cd\u8bbe\u8ba1\u7b56\u7565\u5c31\u7528\u5728\u4e00\u6b65&#xff1b;<\/li>\n<li>\u63cf\u8ff0\u7b97\u6cd5&#xff1a;\u7528\u81ea\u7136\u8bed\u8a00\/\u6d41\u7a0b\u56fe\/\u4f2a\u4ee3\u7801\/\u7a0b\u5e8f\u8bed\u8a00\u628a\u7b97\u6cd5\u5199\u51fa\u6765&#xff1b;<\/li>\n<li>\u8bc1\u660e\u7b97\u6cd5\u6b63\u786e\u6027&#xff1a;\u8bba\u8bc1\u5bf9\u4e00\u5207\u5408\u6cd5\u8f93\u5165\u7b97\u6cd5\u90fd\u7ed9\u51fa\u6b63\u786e\u8f93\u51fa&#xff08;\u6570\u5b66\u5f52\u7eb3\u6cd5\u662f\u5e38\u7528\u5de5\u5177&#xff09;&#xff1b;<\/li>\n<li>\u7b97\u6cd5\u5206\u6790&#xff1a;\u8ba1\u7b97\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6&#xff0c;\u8bc4\u4ef7\u7b97\u6cd5\u4f18\u52a3&#xff0c;\u5fc5\u8981\u65f6\u56de\u5230\u7b2c2\u6b65\u6539\u8fdb\u3002<\/li>\n<hr \/>\n<h3>\u5341\u4e8c\u30011.2 \u7b97\u6cd5\u7684\u5206\u6790&#xff1a;\u65f6\u95f4\u590d\u6742\u5ea6\u6982\u8ff0<\/h3>\n<p>\u4ece\u8fd9\u91cc\u5f00\u59cb\u8fdb\u5165\u672c\u7ae0\u7684\u6838\u5fc3\u8ba1\u7b97\u90e8\u5206\u2014\u2014\u7b97\u6cd5\u5206\u6790\u3002<\/p>\n<h4>12.1 \u4ec0\u4e48\u662f\u7b97\u6cd5\u5206\u6790<\/h4>\n<p>\u7b97\u6cd5\u5206\u6790\u662f\u5206\u6790\u7b97\u6cd5\u5360\u7528\u8ba1\u7b97\u673a\u8d44\u6e90\u7684\u60c5\u51b5&#xff0c;\u4e3b\u8981\u662f\u5206\u6790\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n<ul>\n<li>\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;\u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u968f\u95ee\u9898\u89c4\u6a21 n \u589e\u957f\u7684\u6e10\u8fdb\u8d8b\u52bf&#xff1b;<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;\u7b97\u6cd5\u5360\u7528\u5b58\u50a8\u7a7a\u95f4\u968f\u95ee\u9898\u89c4\u6a21 n \u589e\u957f\u7684\u6e10\u8fdb\u8d8b\u52bf\u3002<\/li>\n<\/ul>\n<p>\u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u6d4b\u91cf&#034;\u8fd0\u884c\u4e86\u591a\u5c11\u79d2&#034;&#xff1f;\u56e0\u4e3a\u5b9e\u9645\u8fd0\u884c\u65f6\u95f4\u53d7\u673a\u5668\u6027\u80fd\u3001\u7f16\u8bd1\u5668\u4f18\u5316\u3001\u64cd\u4f5c\u7cfb\u7edf\u8c03\u5ea6\u7b49\u592a\u591a\u56e0\u7d20\u5f71\u54cd&#xff0c;\u540c\u4e00\u7b97\u6cd5\u5728\u4e0d\u540c\u673a\u5668\u4e0a\u65f6\u95f4\u4e0d\u540c\u3002\u7b97\u6cd5\u5206\u6790\u8981\u7684\u662f\u4e0e\u673a\u5668\u65e0\u5173\u7684\u3001\u672c\u8d28\u6027\u7684\u5ea6\u91cf\u2014\u2014\u8fd9\u5c31\u662f\u6e10\u8fdb\u5206\u6790&#xff08;Asymptotic Analysis&#xff09;\u7684\u601d\u60f3&#xff1a;\u53ea\u5173\u5fc3\u5f53\u95ee\u9898\u89c4\u6a21 n \u8d8b\u5411\u65e0\u7a77\u5927\u65f6&#xff0c;\u8fd0\u884c\u65f6\u95f4\u7684\u589e\u957f\u9636\u3002<\/p>\n<h4>12.2 \u7b97\u6cd5\u8fd0\u884c\u65f6\u95f4\u7684\u6784\u6210<\/h4>\n<p>\u7b97\u6cd5\u662f\u7531\u63a7\u5236\u7ed3\u6784&#xff08;\u987a\u5e8f\u3001\u5206\u652f\u548c\u5faa\u73af3\u79cd&#xff09;\u548c\u539f\u64cd\u4f5c&#xff08;\u6307\u56fa\u6709\u6570\u636e\u7c7b\u578b\u7684\u64cd\u4f5c&#xff09;\u6784\u6210\u7684&#xff0c;\u8fd0\u884c\u65f6\u95f4\u53d6\u51b3\u4e8e\u4e24\u8005\u7684\u7efc\u5408\u6548\u679c\u3002<\/p>\n<ul>\n<li>\u63a7\u5236\u7ed3\u6784&#xff1a;\u987a\u5e8f\u7ed3\u6784&#xff08;\u4f9d\u6b21\u6267\u884c&#xff09;\u3001\u5206\u652f\u7ed3\u6784&#xff08;if\/switch \u9009\u62e9\u6267\u884c&#xff09;\u3001\u5faa\u73af\u7ed3\u6784&#xff08;for\/while \u91cd\u590d\u6267\u884c&#xff09;\u2014\u2014\u5176\u4e2d\u5faa\u73af\u7ed3\u6784\u5bf9\u8fd0\u884c\u65f6\u95f4\u7684\u5f71\u54cd\u6700\u5927&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790\u7684\u4e3b\u6218\u573a\u5c31\u662f\u6570\u5faa\u73af\u6b21\u6570&#xff1b;<\/li>\n<li>\u539f\u64cd\u4f5c&#xff1a;\u56fa\u6709\u6570\u636e\u7c7b\u578b\u7684\u64cd\u4f5c&#xff0c;\u5982\u6574\u6570\u7684\u52a0\u51cf\u4e58\u9664\u3001\u6bd4\u8f83\u3001\u8d4b\u503c\u7b49&#xff0c;\u6bcf\u4e2a\u539f\u64cd\u4f5c\u7684\u6267\u884c\u65f6\u95f4\u662f\u5e38\u6570\u7ea7\u7684\u3002<\/li>\n<\/ul>\n<p>\u770b\u6559\u6750\u4e2d\u7684\u4f8b\u5b50&#xff1a;<\/p>\n<p>bool <span class=\"token function\">Solve<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">double<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>MAX<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> m<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">double<\/span> <span class=\"token operator\">&amp;<\/span>s<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n    s <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>m <span class=\"token operator\">!&#061;<\/span> n<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">return<\/span> false<span class=\"token punctuation\">;<\/span>      <span class=\"token comment\">\/\/ \u5206\u652f\u7ed3\u6784&#xff1a;\u975e\u65b9\u9635\u76f4\u63a5\u5931\u8d25<\/span><\/p>\n<p>    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> m<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span>        <span class=\"token comment\">\/\/ \u5faa\u73af\u7ed3\u6784&#xff1a;\u6267\u884cm\u6b21<\/span><br \/>\n        s <span class=\"token operator\">&#043;&#061;<\/span> a<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>              <span class=\"token comment\">\/\/ \u539f\u64cd\u4f5c&#xff1a;\u7d2f\u52a0\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20<\/span><\/p>\n<p>    <span class=\"token keyword\">return<\/span> true<span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u8fd9\u4e2a\u51fd\u6570\u8ba1\u7b97 m\u00d7n \u77e9\u9635\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e4b\u548c&#xff1a;\u5148\u68c0\u67e5\u662f\u5426\u4e3a\u65b9\u9635&#xff08;m\u2260n \u5219\u5931\u8d25&#xff09;&#xff0c;\u7136\u540e\u5faa\u73af m \u6b21\u7d2f\u52a0 a[i][i]\u3002\u5b83\u7684\u8fd0\u884c\u65f6\u95f4\u7531&#034;1\u6b21\u6bd4\u8f83 &#043; m\u6b21\u5faa\u73af&#xff08;\u6bcf\u6b21\u4e00\u4e2a\u52a0\u6cd5&#xff09;&#034;\u6784\u6210&#xff0c;\u5373\u4e0e m \u6210\u6b63\u6bd4&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(m)\u3002<\/p>\n<h4>12.3 \u95ee\u9898\u89c4\u6a21\u4e0e\u57fa\u672c\u8bed\u53e5<\/h4>\n<ul>\n<li>\u95ee\u9898\u89c4\u6a21 n&#xff1a;\u6307\u8f93\u5165\u6570\u636e\u91cf\u7684\u5927\u5c0f&#xff0c;\u5982\u6570\u7ec4\u5143\u7d20\u4e2a\u6570\u3001\u77e9\u9635\u9636\u6570\u3001\u56fe\u4e2d\u9876\u70b9\u6570\u7b49&#xff1b;<\/li>\n<li>\u57fa\u672c\u8bed\u53e5&#xff08;\u57fa\u672c\u64cd\u4f5c&#xff09;&#xff1a;\u7b97\u6cd5\u4e2d\u6267\u884c\u6b21\u6570\u6700\u591a\u3001\u5bf9\u8fd0\u884c\u65f6\u95f4\u8d77\u51b3\u5b9a\u6027\u4f5c\u7528\u7684\u8bed\u53e5&#xff0c;\u901a\u5e38\u662f\u6700\u6df1\u5c42\u5faa\u73af\u4f53\u5185\u7684\u8bed\u53e5\u3002<\/li>\n<\/ul>\n<p>\u5206\u6790\u65f6\u95f4\u590d\u6742\u5ea6&#xff0c;\u672c\u8d28\u4e0a\u5c31\u662f&#xff1a;\u6c42\u51fa\u57fa\u672c\u8bed\u53e5\u7684\u6267\u884c\u6b21\u6570 f(n)&#xff0c;\u518d\u7528\u6e10\u8fdb\u7b26\u53f7\u8868\u793a f(n) \u7684\u9636\u3002<\/p>\n<h4>12.4 \u5206\u6790\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u4e00\u822c\u6b65\u9aa4<\/h4>\n<p>\u6559\u6750\u7ed9\u51fa\u7684\u6807\u51c6\u6d41\u7a0b&#xff1a;<\/p>\n<p>\u7b97\u6cd5<br \/>\n  \u2193<br \/>\n\u5206\u6790\u95ee\u9898\u89c4\u6a21n&#xff0c;\u627e\u51fa\u57fa\u672c\u8bed\u53e5&#xff0c;\u6c42\u51fa\u5176\u8fd0\u884c\u6b21\u6570 f(n)<br \/>\n  \u2193<br \/>\n\u7528 O\u3001\u03a9 \u6216 \u0398 \u8868\u793a\u5176\u9636<br \/>\n&#xff08;\u4e0a\u754c\u3001\u4e0b\u754c\u3001\u51c6\u786e\u754c&#xff09;<\/p>\n<p>\u5177\u4f53\u64cd\u4f5c\u53e3\u8bc0&#xff08;\u7b14\u8005\u603b\u7ed3&#xff09;&#xff1a;<\/p>\n<li>\u627e&#xff1a;\u627e\u5230\u7b97\u6cd5\u4e2d\u6700\u6df1\u5c42\u7684\u5faa\u73af&#xff0c;\u786e\u5b9a\u57fa\u672c\u8bed\u53e5&#xff1b;<\/li>\n<li>\u6570&#xff1a;\u8ba1\u7b97\u57fa\u672c\u8bed\u53e5\u7684\u6267\u884c\u6b21\u6570&#xff0c;\u5f97\u5230\u5173\u4e8e n \u7684\u8868\u8fbe\u5f0f f(n)&#xff08;\u53ef\u80fd\u9700\u8981\u7528\u5230\u6c42\u548c\u516c\u5f0f&#xff09;&#xff1b;<\/li>\n<li>\u5316&#xff1a;\u53ea\u4fdd\u7559 f(n) \u7684\u6700\u9ad8\u9636\u9879&#xff0c;\u5e76\u53bb\u6389\u7cfb\u6570&#xff0c;\u7528\u5927O\u8868\u793a&#xff0c;\u5373\u5f97\u65f6\u95f4\u590d\u6742\u5ea6\u3002<\/li>\n<p>\u5e38\u7528\u6c42\u548c\u516c\u5f0f&#xff08;\u5fc5\u987b\u80cc\u719f&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>         i<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>         ,<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>          i<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           (<\/p>\n<p>           2<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          6<\/p>\n<p>         ,<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>          k<\/p>\n<p>          2<\/p>\n<p>          i<\/p>\n<p>         &#061;<\/p>\n<p>          2<\/p>\n<p>           k<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>        \\\\sum_{i&#061;1}^{n} i &#061; \\\\frac{n(n&#043;1)}{2}, \\\\qquad \\\\sum_{i&#061;1}^{n} i^2 &#061; \\\\frac{n(n&#043;1)(2n&#043;1)}{6}, \\\\qquad \\\\sum_{i&#061;0}^{k} 2^i &#061; 2^{k&#043;1}-1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.9291em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.9291em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 2em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.1138em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">6<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 2em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8361em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8747em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9824em;vertical-align: -0.0833em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<hr \/>\n<h3>\u5341\u4e09\u3001\u6e10\u8fdb\u7b26\u53f7&#xff1a;\u5927O\u3001\u5927\u03a9\u3001\u5927\u0398 \u8be6\u89e3<\/h3>\n<p>\u6e10\u8fdb\u7b26\u53f7\u662f\u7b97\u6cd5\u5206\u6790\u7684\u6570\u5b66\u8bed\u8a00&#xff0c;\u5171\u4e09\u79cd&#xff1a;\u5927O&#xff08;\u4e0a\u754c&#xff09;\u3001\u5927\u03a9&#xff08;\u4e0b\u754c&#xff09;\u3001\u5927\u0398&#xff08;\u7d27\u786e\u754c&#xff09;\u3002\u8bbe n \u4e3a\u7b97\u6cd5\u4e2d\u7684\u95ee\u9898\u89c4\u6a21&#xff0c;\u901a\u5e38\u7528\u5927O\u3001\u5927\u03a9&#xff08;Omega&#xff09;\u548c\u0398&#xff08;Theta&#xff09;\u4e09\u79cd\u6e10\u8fdb\u7b26\u53f7\u8868\u793a\u7b97\u6cd5\u7684\u6267\u884c\u65f6\u95f4\u4e0e n \u4e4b\u95f4\u7684\u4e00\u79cd\u589e\u957f\u5173\u7cfb\u3002<\/p>\n<h4>13.1 \u5b9a\u4e491&#xff1a;\u5927O\u7b26\u53f7&#xff08;\u4e0a\u754c&#xff09;<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; O(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span>&#xff08;\u8bfb\u4f5c&#034;f(n)\u662fg(n)\u7684\u5927O&#034;&#xff09;\u5f53\u4e14\u4ec5\u5f53\u5b58\u5728\u6b63\u5e38\u91cf c \u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          n<\/p>\n<p>          0<\/p>\n<p>        n_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4f7f\u5f97\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         n<\/p>\n<p>         \u2265<\/p>\n<p>          n<\/p>\n<p>          0<\/p>\n<p>        n \\\\ge n_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         \u2264<\/p>\n<p>         c<\/p>\n<p>         \u22c5<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        f(n) \\\\le c \\\\cdot g(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 g(n) \u4e3a f(n) \u7684\u4e0a\u754c\u3002<\/p>\n<p>\u76f4\u89c2\u7406\u89e3&#xff1a;\u53ea\u8981 n \u8db3\u591f\u5927&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \u2265<\/p>\n<p>         n<\/p>\n<p>         0<\/p>\n<p>       n \\\\ge n_0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;f(n) \u7684\u56fe\u50cf\u603b\u88ab c\u00b7g(n) \u7684\u56fe\u50cf&#034;\u538b&#034;\u5728\u4e0b\u9762\u3002\u4e5f\u5c31\u662f\u8bf4&#xff0c;f(n) \u7684\u589e\u957f\u6700\u591a\u50cf g(n) \u90a3\u6837\u5feb\u3002<\/p>\n<p>\u4f8b\u5b501&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       3n&#043;2 &#061; O(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002 \u8bc1\u660e\u601d\u8def&#xff1a;\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \u2265<\/p>\n<p>        2<\/p>\n<p>       n \\\\ge 2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        \u2264<\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        n<\/p>\n<p>        &#061;<\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>       3n&#043;2 \\\\le 3n&#043;n &#061; 4n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>\u3002\u53d6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        c<\/p>\n<p>        &#061;<\/p>\n<p>        4<\/p>\n<p>        ,<\/p>\n<p>         n<\/p>\n<p>         0<\/p>\n<p>        &#061;<\/p>\n<p>        2<\/p>\n<p>       c&#061;4, n_0&#061;2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">4<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span> \u5373\u53ef\u3002\u2705<\/p>\n<p>\u4f8b\u5b502&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>         n<\/p>\n<p>         4<\/p>\n<p>        )<\/p>\n<p>       10n^2&#043;4n&#043;2 &#061; O(n^4)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002 \u8bc1\u660e\u601d\u8def&#xff1a;\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \u2265<\/p>\n<p>        2<\/p>\n<p>       n \\\\ge 2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        \u2264<\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         4<\/p>\n<p>       10n^2&#043;4n&#043;2 \\\\le 10n^4<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;\u56e0\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        \u2264<\/p>\n<p>         n<\/p>\n<p>         4<\/p>\n<p>       n^2 \\\\le n^4<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9501em;vertical-align: -0.136em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>        \u2264<\/p>\n<p>         n<\/p>\n<p>         4<\/p>\n<p>       4n \\\\le n^4<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        2<\/p>\n<p>        \u2264<\/p>\n<p>         n<\/p>\n<p>         4<\/p>\n<p>       2 \\\\le n^4<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5747\u6210\u7acb&#xff0c;\u52a0\u8d77\u6765\u4e0d\u8d85\u8fc7 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         4<\/p>\n<p>        &#043;<\/p>\n<p>        .<\/p>\n<p>        .<\/p>\n<p>        .<\/p>\n<p>       10n^4&#043;&#8230;<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.1056em\"><\/span><span class=\"mord\">&#8230;<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6559\u6750\u76f4\u63a5\u7ed9\u51fa\u8be5\u653e\u7f29&#xff09;\u3002\u53d6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        c<\/p>\n<p>        &#061;<\/p>\n<p>        10<\/p>\n<p>        ,<\/p>\n<p>         n<\/p>\n<p>         0<\/p>\n<p>        &#061;<\/p>\n<p>        2<\/p>\n<p>       c&#061;10, n_0&#061;2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">10<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span> \u5373\u53ef\u3002\u2705<\/p>\n<p>\u5173\u4e8e&#034;\u7d27\u51d1\u4e0a\u754c&#034;&#xff08;\u7d27\u786e\u4e0a\u754c&#xff09;&#xff1a;<\/p>\n<p>\u5927O\u7b26\u53f7\u7528\u6765\u63cf\u8ff0\u589e\u957f\u7387\u7684\u4e0a\u754c&#xff0c;\u8868\u793a f(n) \u7684\u589e\u957f\u6700\u591a\u50cf g(n) \u589e\u957f\u7684\u90a3\u6837\u5feb&#xff0c;\u5373\u5f53\u8f93\u5165\u89c4\u6a21\u4e3a n \u65f6&#xff0c;\u7b97\u6cd5\u6d88\u8017\u65f6\u95f4\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e2a\u4e0a\u754c\u7684\u9636\u8d8a\u4f4e&#xff0c;\u7ed3\u679c\u5c31\u8d8a\u6709\u4ef7\u503c\u3002\u6240\u4ee5\u5bf9\u4e8e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         &#043;<\/p>\n<p>         4<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         2<\/p>\n<p>        10n^2&#043;4n&#043;2<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         )<\/p>\n<p>        O(n^2)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6bd4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          4<\/p>\n<p>         )<\/p>\n<p>        O(n^4)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6709\u4ef7\u503c\u3002<\/p>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u7684\u65f6\u95f4\u7528\u5927O\u7b26\u53f7\u8868\u793a\u65f6&#xff0c;\u603b\u662f\u91c7\u7528\u6700\u6709\u4ef7\u503c\u7684 g(n) \u8868\u793a&#xff0c;\u79f0\u4e4b\u4e3a \u201c\u7d27\u51d1\u4e0a\u754c\u201d \u6216 \u201c\u7d27\u786e\u4e0a\u754c\u201d \u3002<\/p>\n<p>\u591a\u9879\u5f0f\u7684\u901a\u7528\u7ed3\u8bba&#xff1a;<\/p>\n<p>\u4e00\u822c\u5730&#xff0c;\u5982\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          a<\/p>\n<p>          m<\/p>\n<p>          n<\/p>\n<p>          m<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>           m<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>           m<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>         &#043;<\/p>\n<p>         \u22ef<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>          1<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>          0<\/p>\n<p>        f(n) &#061; a_m n^m &#043; a_{m-1} n^{m-1} &#043; \\\\cdots &#043; a_1 n &#043; a_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8144em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0224em;vertical-align: -0.2083em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          a<\/p>\n<p>          m<\/p>\n<p>         &gt;<\/p>\n<p>         0<\/p>\n<p>        a_m &gt; 0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6891em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          m<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; O(n^m)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u63a8\u5bfc\u8bb0\u5fc6\u6cd5&#xff1a;\u591a\u9879\u5f0f\u7684\u6e10\u8fdb\u9636 &#061; \u6700\u9ad8\u6b21\u9879\u7684\u9636&#xff0c;\u7cfb\u6570\u548c\u4f4e\u6b21\u9879\u5168\u90e8\u6254\u6389\u3002\u4f8b\u5982 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        5<\/p>\n<p>         n<\/p>\n<p>         3<\/p>\n<p>        &#043;<\/p>\n<p>        100<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        7<\/p>\n<p>       f(n)&#061;5n^3&#043;100n^2&#043;n&#043;7<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">5<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">100<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">7<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>         n<\/p>\n<p>         3<\/p>\n<p>        )<\/p>\n<p>       f(n)&#061;O(n^3)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>13.2 \u5b9a\u4e492&#xff1a;\u5927\u03a9\u7b26\u53f7&#xff08;\u4e0b\u754c&#xff09;<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Omega(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span>&#xff08;\u8bfb\u4f5c&#034;f(n)\u662fg(n)\u7684\u5927\u03a9&#034;&#xff09;\u5f53\u4e14\u4ec5\u5f53\u5b58\u5728\u6b63\u5e38\u91cf c \u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          n<\/p>\n<p>          0<\/p>\n<p>        n_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4f7f\u5f97\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         n<\/p>\n<p>         \u2265<\/p>\n<p>          n<\/p>\n<p>          0<\/p>\n<p>        n \\\\ge n_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         \u2265<\/p>\n<p>         c<\/p>\n<p>         \u22c5<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        f(n) \\\\ge c \\\\cdot g(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 g(n) \u4e3a f(n) \u7684\u4e0b\u754c\u3002<\/p>\n<p>\u76f4\u89c2\u7406\u89e3&#xff1a;\u53ea\u8981 n \u8db3\u591f\u5927&#xff0c;f(n) \u7684\u56fe\u50cf\u603b\u88ab c\u00b7g(n) &#034;\u6258&#034;\u5728\u4e0a\u9762\u3002f(n) \u7684\u589e\u957f\u81f3\u5c11\u50cf g(n) \u90a3\u6837\u5feb\u3002<\/p>\n<p>\u4f8b\u5b501&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        &#061;<\/p>\n<p>        \u03a9<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       3n&#043;2 &#061; \\\\Omega(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002 \u8bc1\u660e\u601d\u8def&#xff1a;\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \u2265<\/p>\n<p>        1<\/p>\n<p>       n \\\\ge 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        \u2265<\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>       3n&#043;2 \\\\ge 3n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>\u3002\u53d6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        c<\/p>\n<p>        &#061;<\/p>\n<p>        3<\/p>\n<p>        ,<\/p>\n<p>         n<\/p>\n<p>         0<\/p>\n<p>        &#061;<\/p>\n<p>        1<\/p>\n<p>       c&#061;3, n_0&#061;1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">3<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u5373\u53ef\u3002\u2705<\/p>\n<p>\u4f8b\u5b502&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        &#061;<\/p>\n<p>        \u03a9<\/p>\n<p>        (<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        )<\/p>\n<p>       10n^2&#043;4n&#043;2 &#061; \\\\Omega(n^2)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002 \u8bc1\u660e\u601d\u8def&#xff1a;\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \u2265<\/p>\n<p>        1<\/p>\n<p>       n \\\\ge 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        \u2265<\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        \u2265<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>       10n^2&#043;4n&#043;2 \\\\ge 10n^2 \\\\ge n^2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9501em;vertical-align: -0.136em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u53d6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        c<\/p>\n<p>        &#061;<\/p>\n<p>        1<\/p>\n<p>        ,<\/p>\n<p>         n<\/p>\n<p>         0<\/p>\n<p>        &#061;<\/p>\n<p>        1<\/p>\n<p>       c&#061;1, n_0&#061;1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8389em;vertical-align: -0.1944em\"><\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u5373\u53ef\u3002\u2705<\/p>\n<p>\u5173\u4e8e&#034;\u7d27\u51d1\u4e0b\u754c&#034;&#xff08;\u7d27\u786e\u4e0b\u754c&#xff09;&#xff1a;<\/p>\n<p>\u5927\u03a9\u7b26\u53f7\u7528\u6765\u63cf\u8ff0\u589e\u957f\u7387\u7684\u4e0b\u754c&#xff0c;\u8868\u793a f(n) \u7684\u589e\u957f\u6700\u5c11\u50cf g(n) \u589e\u957f\u7684\u90a3\u6837\u5feb&#xff0c;\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u5f53\u8f93\u5165\u89c4\u6a21\u4e3a n \u65f6&#xff0c;\u7b97\u6cd5\u6d88\u8017\u65f6\u95f4\u7684\u6700\u5c0f\u503c\u3002<\/p>\n<p>\u4e0e\u5927O\u7b26\u53f7\u5bf9\u79f0&#xff1a;\u8fd9\u4e2a\u4e0b\u754c\u7684\u9636\u8d8a\u9ad8&#xff0c;\u7ed3\u679c\u5c31\u8d8a\u6709\u4ef7\u503c\u3002\u6240\u4ee5\u5bf9\u4e8e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         &#043;<\/p>\n<p>         4<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         2<\/p>\n<p>        10n^2&#043;4n&#043;2<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         )<\/p>\n<p>        \\\\Omega(n^2)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6bd4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        \\\\Omega(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6709\u4ef7\u503c\u3002\u4e00\u4e2a\u7b97\u6cd5\u7684\u65f6\u95f4\u7528\u5927\u03a9\u7b26\u53f7\u8868\u793a\u65f6&#xff0c;\u603b\u662f\u91c7\u7528\u6700\u6709\u4ef7\u503c\u7684 g(n) \u8868\u793a&#xff0c;\u79f0\u4e4b\u4e3a \u201c\u7d27\u51d1\u4e0b\u754c\u201d \u6216 \u201c\u7d27\u786e\u4e0b\u754c\u201d\u3002<\/p>\n<p>\u26a0\ufe0f \u6ce8\u610f\u65b9\u5411\u76f8\u53cd&#xff1a;\u5927O\u8981&#034;\u8d8a\u4f4e\u8d8a\u6709\u4ef7\u503c&#034;&#xff08;\u4e0a\u754c\u8d8a\u4f4e\u8bf4\u660e\u5361\u5f97\u8d8a\u7d27&#xff09;&#xff0c;\u5927\u03a9\u8981&#034;\u8d8a\u9ad8\u8d8a\u6709\u4ef7\u503c&#034;&#xff08;\u4e0b\u754c\u8d8a\u9ad8\u8bf4\u660e\u5361\u5f97\u8d8a\u7d27&#xff09;\u3002\u53ef\u4ee5\u60f3\u8c61\u4e24\u4e2a\u5939\u5b50\u4ece\u4e0a\u4e0b\u4e24\u8fb9\u5939 f(n)&#xff1a;\u5939\u5f97\u8d8a\u7d27&#xff0c;\u4fe1\u606f\u91cf\u8d8a\u5927\u3002<\/p>\n<p>\u591a\u9879\u5f0f\u7684\u901a\u7528\u7ed3\u8bba&#xff1a;<\/p>\n<p>\u4e00\u822c\u5730&#xff0c;\u5982\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          a<\/p>\n<p>          m<\/p>\n<p>          n<\/p>\n<p>          m<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>           m<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>           m<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>         &#043;<\/p>\n<p>         \u22ef<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>          1<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>          0<\/p>\n<p>        f(n) &#061; a_m n^m &#043; a_{m-1} n^{m-1} &#043; \\\\cdots &#043; a_1 n &#043; a_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8144em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0224em;vertical-align: -0.2083em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          a<\/p>\n<p>          m<\/p>\n<p>         &gt;<\/p>\n<p>         0<\/p>\n<p>        a_m &gt; 0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6891em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          m<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Omega(n^m)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>13.3 \u5b9a\u4e493&#xff1a;\u5927\u0398\u7b26\u53f7&#xff08;\u540c\u9636\/\u7d27\u786e\u754c&#xff09;<\/h4>\n<p><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Theta(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span>&#xff08;\u8bfb\u4f5c&#034;f(n)\u662fg(n)\u7684\u5927\u0398&#034;&#xff09;\u5f53\u4e14\u4ec5\u5f53\u5b58\u5728\u6b63\u5e38\u91cf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          c<\/p>\n<p>          1<\/p>\n<p>        c_1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          c<\/p>\n<p>          2<\/p>\n<p>        c_2<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          n<\/p>\n<p>          0<\/p>\n<p>        n_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4f7f\u5f97\u5f53 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         n<\/p>\n<p>         \u2265<\/p>\n<p>          n<\/p>\n<p>          0<\/p>\n<p>        n \\\\ge n_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          c<\/p>\n<p>          1<\/p>\n<p>         \u22c5<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         \u2264<\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         \u2264<\/p>\n<p>          c<\/p>\n<p>          2<\/p>\n<p>         \u22c5<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        c_1 \\\\cdot g(n) \\\\le f(n) \\\\le c_2 \\\\cdot g(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5945em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5945em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 g(n) \u4e0e f(n) \u540c\u9636\u3002<\/p>\n<p>\u76f4\u89c2\u7406\u89e3&#xff1a;f(n) \u88ab\u4e0a\u4e0b\u4e24\u6761 c\u00b7g(n) \u66f2\u7ebf\u5939\u5728\u4e2d\u95f4\u2014\u2014\u589e\u957f\u65e2\u4e0d\u6bd4 g(n) \u5feb&#xff0c;\u4e5f\u4e0d\u6bd4 g(n) \u6162&#xff0c;\u4e8c\u8005&#034;\u540c\u901f\u589e\u957f&#034;\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        3<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        &#061;<\/p>\n<p>        \u0398<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       3n&#043;2 &#061; \\\\Theta(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff1b;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        10<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        4<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        2<\/p>\n<p>        &#061;<\/p>\n<p>        \u0398<\/p>\n<p>        (<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        )<\/p>\n<p>       10n^2&#043;4n&#043;2 &#061; \\\\Theta(n^2)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">10<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u591a\u9879\u5f0f\u7684\u901a\u7528\u7ed3\u8bba&#xff1a;<\/p>\n<p>\u4e00\u822c\u5730&#xff0c;\u5982\u679c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          a<\/p>\n<p>          m<\/p>\n<p>          n<\/p>\n<p>          m<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>           m<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>           m<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>         &#043;<\/p>\n<p>         \u22ef<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>          1<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>          a<\/p>\n<p>          0<\/p>\n<p>        f(n) &#061; a_m n^m &#043; a_{m-1} n^{m-1} &#043; \\\\cdots &#043; a_1 n &#043; a_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8144em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0224em;vertical-align: -0.2083em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          a<\/p>\n<p>          m<\/p>\n<p>         &gt;<\/p>\n<p>         0<\/p>\n<p>        a_m &gt; 0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6891em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">a<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          m<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Theta(n^m)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u4e09\u8005\u7684\u5173\u7cfb&#xff1a;<\/p>\n<p>\u5927\u0398\u7b26\u53f7\u6bd4\u5927O\u7b26\u53f7\u548c\u5927\u03a9\u7b26\u53f7\u90fd\u7cbe\u786e\u3002<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Theta(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span> \u5f53\u4e14\u4ec5\u5f53 g(n) \u65e2\u662f f(n) \u7684\u4e0a\u754c&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        f(n)&#061;O(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span>&#xff09;\u53c8\u662f f(n) \u7684\u4e0b\u754c&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        f(n)&#061;\\\\Omega(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span>&#xff09;\u3002<\/p>\n<p>\u5373&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u0398<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        \u2229<\/p>\n<p>        \u03a9<\/p>\n<p>       \\\\Theta &#061; O \\\\cap \\\\Omega<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2229<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord\">\u03a9<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>13.4 \u4e09\u79cd\u6e10\u8fdb\u7b26\u53f7\u5bf9\u6bd4\u8868&#xff08;\u80cc\u8bf5\u7248&#xff09;<\/h4>\n<table>\n<tr>\u7b26\u53f7\u540d\u79f0\u6570\u5b66\u5b9a\u4e49&#xff08;n\u2265n\u2080\u65f6&#xff09;\u76f4\u89c2\u542b\u4e49\u754c\u7684\u65b9\u5411\u4ef7\u503c\u53d6\u5411<\/tr>\n<tbody>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>          O<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5927O<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           f<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           \u2264<\/p>\n<p>           c<br \/>\n          \u2009<\/p>\n<p>           g<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          f(n) \\\\le c\\\\,g(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u589e\u957f\u7387\u4e0a\u754c&#xff0c;\u6700\u591a\u8fd9\u4e48\u5feb<\/td>\n<td>\u4e0a\u754c<\/td>\n<td>\u9636\u8d8a\u4f4e\u8d8a\u6709\u4ef7\u503c&#xff08;\u7d27\u51d1\u4e0a\u754c&#xff09;<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           \u03a9<\/p>\n<p>          \\\\Omega<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord\">\u03a9<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5927\u03a9<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           f<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           \u2265<\/p>\n<p>           c<br \/>\n          \u2009<\/p>\n<p>           g<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          f(n) \\\\ge c\\\\,g(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2265<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u589e\u957f\u7387\u4e0b\u754c&#xff0c;\u81f3\u5c11\u8fd9\u4e48\u5feb<\/td>\n<td>\u4e0b\u754c<\/td>\n<td>\u9636\u8d8a\u9ad8\u8d8a\u6709\u4ef7\u503c&#xff08;\u7d27\u51d1\u4e0b\u754c&#xff09;<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           \u0398<\/p>\n<p>          \\\\Theta<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord\">\u0398<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5927\u0398<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>            c<\/p>\n<p>            1<\/p>\n<p>           g<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           \u2264<\/p>\n<p>           f<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           \u2264<\/p>\n<p>            c<\/p>\n<p>            2<\/p>\n<p>           g<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          c_1 g(n) \\\\le f(n) \\\\le c_2 g(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">c<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u540c\u9636&#xff0c;\u4e0d\u5feb\u4e0d\u6162\u521a\u521a\u597d<\/td>\n<td>\u4e0a\u4e0b\u5939\u903c<\/td>\n<td>\u6700\u7cbe\u786e&#xff0c;O\u4e0e\u03a9\u540c\u65f6\u6210\u7acb<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>13.5 \u5e38\u89c1\u51fd\u6570\u589e\u957f\u9636\u6392\u5e8f&#xff08;\u5fc5\u987b\u70c2\u719f\u4e8e\u5fc3&#xff09;<\/h4>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          3<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          2<\/p>\n<p>          n<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         !<\/p>\n<p>         )<\/p>\n<p>         &lt;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          n<\/p>\n<p>         )<\/p>\n<p>        O(1) &lt; O(\\\\log n) &lt; O(n) &lt; O(n\\\\log n) &lt; O(n^2) &lt; O(n^3) &lt; O(2^n) &lt; O(n!) &lt; O(n^n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7144em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7144em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<table>\n<tr>\u590d\u6742\u5ea6\u540d\u79f0\u5178\u578b\u7b97\u6cd5\u4e3e\u4f8bn&#061;10\u2076 \u65f6\u7684\u611f\u53d7<\/tr>\n<tbody>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          O(1)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5e38\u6570\u9636<\/td>\n<td>\u6570\u7ec4\u6309\u4e0b\u6807\u8bbf\u95ee\u3001\u5165\u6808\u51fa\u6808<\/td>\n<td>\u77ac\u95f4<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(\\\\log n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5bf9\u6570\u9636<\/td>\n<td>\u4e8c\u5206\u67e5\u627e<\/td>\n<td>\u7ea620\u6b21\u64cd\u4f5c&#xff0c;\u77ac\u95f4<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u7ebf\u6027\u9636<\/td>\n<td>\u987a\u5e8f\u67e5\u627e\u3001\u904d\u5386\u6570\u7ec4<\/td>\n<td>\u767e\u4e07\u6b21\u64cd\u4f5c&#xff0c;\u6beb\u79d2\u7ea7<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(n\\\\log n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u7ebf\u6027\u5bf9\u6570\u9636<\/td>\n<td>\u5f52\u5e76\u6392\u5e8f\u3001\u5feb\u901f\u6392\u5e8f&#xff08;\u5e73\u5747&#xff09;\u3001\u5806\u6392\u5e8f<\/td>\n<td>\u5343\u4e07\u7ea7\u64cd\u4f5c&#xff0c;\u79d2\u7ea7\u4ee5\u5185<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>            n<\/p>\n<p>            2<\/p>\n<p>           )<\/p>\n<p>          O(n^2)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5e73\u65b9\u9636<\/td>\n<td>\u7b80\u5355\u9009\u62e9\/\u63d2\u5165\/\u5192\u6ce1\u6392\u5e8f\u3001\u53cc\u91cd\u5faa\u73af<\/td>\n<td>\u4e07\u4ebf\u6b21\u64cd\u4f5c&#xff0c;\u5c0f\u65f6\u7ea7&#xff0c;\u4e0d\u53ef\u63a5\u53d7<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>            n<\/p>\n<p>            3<\/p>\n<p>           )<\/p>\n<p>          O(n^3)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u7acb\u65b9\u9636<\/td>\n<td>\u4e09\u91cd\u5faa\u73af\u3001\u6734\u7d20\u77e9\u9635\u4e58\u6cd5<\/td>\n<td>\u5b8c\u5168\u4e0d\u53ef\u63a5\u53d7<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>            2<\/p>\n<p>            n<\/p>\n<p>           )<\/p>\n<p>          O(2^n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u6307\u6570\u9636<\/td>\n<td>\u5b50\u96c6\u679a\u4e3e\u3001\u6734\u7d20\u6590\u6ce2\u90a3\u5951\u9012\u5f52<\/td>\n<td>n&#061;30\u5c31\u7206\u70b8<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           !<\/p>\n<p>           )<\/p>\n<p>          O(n!)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">!)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u9636\u4e58\u9636<\/td>\n<td>\u5168\u6392\u5217\u679a\u4e3e\u3001TSP\u66b4\u529b\u89e3\u6cd5<\/td>\n<td>n&#061;15\u5c31\u7206\u70b8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&#x1f4a1; \u5b9e\u8df5\u610f\u4e49&#xff1a;\u4e00\u822c\u8ba4\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       O(n\\\\log n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u53ca\u4ee5\u4e0b\u662f&#034;\u9ad8\u6548\u7b97\u6cd5&#034;&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        )<\/p>\n<p>       O(n^2)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5728 n \u8f83\u5927\u65f6\u5df2\u7ecf\u5403\u529b&#xff0c;\u6307\u6570\u9636\/\u9636\u4e58\u9636\u7b97\u6cd5\u53ea\u5bf9\u6781\u5c0f\u89c4\u6a21\u95ee\u9898\u53ef\u884c&#xff08;\u8fd9\u7c7b\u95ee\u9898\u5f80\u5f80\u5c5e\u4e8eNP\u96be\u95ee\u9898&#xff0c;\u8fd9\u6b63\u662f\u8bfe\u7a0b\u540e\u9762\u7ae0\u8282\u8981\u8ba8\u8bba\u7684\u5185\u5bb9&#xff09;\u3002<\/p>\n<hr \/>\n<h3>\u5341\u56db\u3001\u3010\u4f8b1-3\u3011\u4e09\u91cd\u5faa\u73af\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790&#xff08;\u542b\u5b8c\u6574\u63a8\u5bfc&#xff09;<\/h3>\n<h4>14.1 \u9898\u76ee<\/h4>\n<p>\u5206\u6790\u4ee5\u4e0b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">fun<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> s <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">,<\/span> k<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>j <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> j <span class=\"token operator\">&lt;&#061;<\/span> i<span class=\"token punctuation\">;<\/span> j<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span> k <span class=\"token operator\">&lt;<\/span> j<span class=\"token punctuation\">;<\/span> k<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                s<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h4>14.2 \u5206\u6790\u8fc7\u7a0b<\/h4>\n<p>\u7b2c\u4e00\u6b65&#xff1a;\u627e\u57fa\u672c\u8bed\u53e5\u3002 \u8be5\u7b97\u6cd5\u7684\u57fa\u672c\u8bed\u53e5\u662f s&#043;&#043;\u2014\u2014\u5b83\u4f4d\u4e8e\u6700\u5185\u5c42\u5faa\u73af\u4f53\u4e2d&#xff0c;\u6267\u884c\u6b21\u6570\u6700\u591a\u3002<\/p>\n<p>\u7b2c\u4e8c\u6b65&#xff1a;\u6570\u6267\u884c\u6b21\u6570 f(n)\u3002 \u4e09\u5c42\u5faa\u73af\u7684\u8fb9\u754c\u76f8\u4e92\u4f9d\u8d56&#xff1a;i \u4ece 0 \u5230 n&#xff1b;\u5bf9\u6bcf\u4e2a i&#xff0c;j \u4ece 0 \u5230 i&#xff1b;\u5bf9\u6bcf\u4e2a j&#xff0c;k \u4ece 0 \u5230 j\u22121\u3002\u56e0\u6b64 s&#043;&#043; \u7684\u603b\u6267\u884c\u6b21\u6570\u4e3a\u4e09\u91cd\u6c42\u548c&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>          n<\/p>\n<p>          \u2211<\/p>\n<p>           j<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>          i<\/p>\n<p>          \u2211<\/p>\n<p>           k<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>           j<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>         1<\/p>\n<p>         f(n)&#061;\\\\sum_{i&#061;0}^{n}\\\\sum_{j&#061;0}^{i}\\\\sum_{k&#061;0}^{j-1}1 <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.2726em;vertical-align: -1.4138em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8117em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8588em\"><span class=\"\" style=\"top: -1.8479em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3471em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3021em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7b2c\u4e09\u6b65&#xff1a;\u7531\u5185\u5411\u5916\u9010\u5c42\u5316\u7b80\u3002<\/p>\n<p>\u6700\u5185\u5c42&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>          k<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>          j<\/p>\n<p>          \u2212<\/p>\n<p>          1<\/p>\n<p>        1<\/p>\n<p>        &#061;<\/p>\n<p>        j<\/p>\n<p>        \u2212<\/p>\n<p>        1<\/p>\n<p>        \u2212<\/p>\n<p>        0<\/p>\n<p>        &#043;<\/p>\n<p>        1<\/p>\n<p>        &#061;<\/p>\n<p>        j<\/p>\n<p>       \\\\sum_{k&#061;0}^{j-1}1 &#061; j-1-0&#043;1 &#061; j<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.2643em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9646em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.854em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">0<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.854em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span>&#xff08;k \u4ece 0 \u6570\u5230 j\u22121&#xff0c;\u5171 j \u6b21&#xff09;\u3002<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>          n<\/p>\n<p>          \u2211<\/p>\n<p>           j<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>          i<\/p>\n<p>         j<\/p>\n<p>         f(n)&#061;\\\\sum_{i&#061;0}^{n}\\\\sum_{j&#061;0}^{i} j <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.2254em;vertical-align: -1.4138em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8117em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4138em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4e2d\u95f4\u5c42\u7528\u6c42\u548c\u516c\u5f0f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>          j<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>         i<\/p>\n<p>        j<\/p>\n<p>        &#061;<\/p>\n<p>           i<\/p>\n<p>           (<\/p>\n<p>           i<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>       \\\\sum_{j&#061;0}^{i} j &#061; \\\\dfrac{i(i&#043;1)}{2}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.4004em;vertical-align: -0.4358em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9646em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4358em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>          n<\/p>\n<p>           i<\/p>\n<p>           (<\/p>\n<p>           i<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>          1<\/p>\n<p>          2<\/p>\n<p>          (<\/p>\n<p>           \u2211<\/p>\n<p>            i<\/p>\n<p>            &#061;<\/p>\n<p>            0<\/p>\n<p>           n<\/p>\n<p>           i<\/p>\n<p>           2<\/p>\n<p>          &#043;<\/p>\n<p>           \u2211<\/p>\n<p>            i<\/p>\n<p>            &#061;<\/p>\n<p>            0<\/p>\n<p>           n<\/p>\n<p>          i<\/p>\n<p>          )<\/p>\n<p>         f(n)&#061;\\\\sum_{i&#061;0}^{n}\\\\frac{i(i&#043;1)}{2}&#061;\\\\frac{1}{2}\\\\left(\\\\sum_{i&#061;0}^{n}i^2&#043;\\\\sum_{i&#061;0}^{n}i\\\\right) <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.9291em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0277em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">(<\/span><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.6514em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4ee3\u5165\u516c\u5f0f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>          i<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>         n<\/p>\n<p>         i<\/p>\n<p>         2<\/p>\n<p>        &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           (<\/p>\n<p>           2<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          6<\/p>\n<p>       \\\\sum_{i&#061;0}^{n} i^2 &#061; \\\\dfrac{n(n&#043;1)(2n&#043;1)}{6}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1138em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">6<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span> \u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>          i<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>         n<\/p>\n<p>        i<\/p>\n<p>        &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>       \\\\sum_{i&#061;0}^{n} i &#061; \\\\dfrac{n(n&#043;1)}{2}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.104em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          1<\/p>\n<p>          2<\/p>\n<p>         \u22c5<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           (<\/p>\n<p>           2<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          6<\/p>\n<p>         &#043;<\/p>\n<p>          1<\/p>\n<p>          2<\/p>\n<p>         \u22c5<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           (<\/p>\n<p>           2<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          12<\/p>\n<p>         &#043;<\/p>\n<p>           3<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          12<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           (<\/p>\n<p>           2<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           4<\/p>\n<p>           )<\/p>\n<p>          12<\/p>\n<p>         &#061;<\/p>\n<p>           2<\/p>\n<p>            n<\/p>\n<p>            3<\/p>\n<p>           &#043;<\/p>\n<p>           6<\/p>\n<p>            n<\/p>\n<p>            2<\/p>\n<p>           &#043;<\/p>\n<p>           4<\/p>\n<p>           n<\/p>\n<p>          12<\/p>\n<p>         f(n)&#061;\\\\frac{1}{2}\\\\cdot\\\\frac{n(n&#043;1)(2n&#043;1)}{6}&#043;\\\\frac{1}{2}\\\\cdot\\\\frac{n(n&#043;1)}{2} &#061;\\\\frac{n(n&#043;1)(2n&#043;1)}{12}&#043;\\\\frac{3n(n&#043;1)}{12} &#061;\\\\frac{n(n&#043;1)(2n&#043;4)}{12} &#061;\\\\frac{2n^3&#043;6n^2&#043;4n}{12} <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">6<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">12<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">12<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">12<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">4<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.1771em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.4911em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">12<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">6<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">4<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7b2c\u56db\u6b65&#xff1a;\u53d6\u9636\u3002 \u53ea\u4fdd\u7559\u6700\u9ad8\u9636\u9879 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        2<\/p>\n<p>         n<\/p>\n<p>         3<\/p>\n<p>        \/<\/p>\n<p>        12<\/p>\n<p>       2n^3\/12<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\">\/12<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u53bb\u6389\u7cfb\u6570\u5f97&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          3<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; O(n^3)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1141em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7ed3\u8bba&#xff1a;\u8be5\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>            n<\/p>\n<p>            3<\/p>\n<p>           )<\/p>\n<p>        \\\\boldsymbol{O(n^3)}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.0319em\">O<\/span><span class=\"mopen mathbf\">(<\/span><span class=\"mord\"><span class=\"mord boldsymbol\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathbf mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose mathbf\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>14.3 \u6280\u5de7\u70b9\u62e8<\/h4>\n<ul>\n<li>\u5176\u5b9e\u5bf9\u4e8e\u8fd9\u79cd\u9898\u76ee&#xff0c;\u4e0d\u5fc5\u7cbe\u786e\u6c42\u548c\u4e5f\u53ef\u4ee5\u5feb\u901f\u5224\u65ad&#xff1a;\u6700\u5185\u5c42\u6267\u884c\u6b21\u6570\u91cf\u7ea7\u4e0d\u8d85\u8fc7 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         \u223c<\/p>\n<p>          n<\/p>\n<p>          3<\/p>\n<p>        (n&#043;1)(n&#043;1)(n&#043;1) \\\\sim n^3<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u223c<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4e14\u5f53 i\u3001j \u53d6\u8f83\u5927\u503c\u65f6\u5185\u5c42\u786e\u5b9e\u6267\u884c \u0398(n) \u6b21&#xff0c;\u6545\u603b\u6b21\u6570\u4e3a \u0398(n\u00b3)\u3002\u4f46\u8003\u8bd5\u4e2d\u82e5\u8981\u6c42&#034;\u5206\u6790&#034;&#xff0c;\u6700\u597d\u5199\u51fa\u6c42\u548c\u63a8\u5bfc\u8fc7\u7a0b&#xff1b;<\/li>\n<li>\u6ce8\u610f\u672c\u9898 j \u5faa\u73af\u662f j &lt;&#061; i\u3001k \u5faa\u73af\u662f k &lt; j&#xff0c;\u8fb9\u754c\u5404\u4e0d\u76f8\u540c&#xff0c;\u9010\u5c42\u5199\u6e05\u6c42\u548c\u4e0a\u4e0b\u9650\u662f\u907f\u514d\u51fa\u9519\u7684\u5173\u952e&#xff1b;<\/li>\n<li>\u5e38\u89c1\u7684\u4e09\u5c42&#034;\u4e09\u89d2\u5f62&#034;\u5faa\u73af&#xff08;i:0..n, j:0..i, k:0..j&#xff09;\u7b54\u6848\u540c\u6837\u662f O(n\u00b3)&#xff0c;\u53ea\u662f\u5e38\u6570\u4e0d\u540c\u2014\u2014\u6e10\u8fdb\u5206\u6790\u4e0d\u5728\u4e4e\u5e38\u6570\u3002<\/li>\n<\/ul>\n<hr \/>\n<h3>\u5341\u4e94\u3001\u7b97\u6cd5\u7684\u6700\u597d\u3001\u6700\u574f\u548c\u5e73\u5747\u60c5\u51b5<\/h3>\n<p>\u540c\u4e00\u4e2a\u7b97\u6cd5&#xff0c;\u9762\u5bf9\u4e0d\u540c\u7684\u8f93\u5165&#xff0c;\u6267\u884c\u65f6\u95f4\u53ef\u80fd\u5927\u76f8\u5f84\u5ead\u3002\u4f8b\u5982\u6392\u5e8f\u7b97\u6cd5&#xff1a;\u8f93\u5165\u5df2\u7ecf\u6709\u5e8f\u65f6\u53ef\u80fd\u53ea\u626b\u4e00\u904d&#xff0c;\u8f93\u5165\u9006\u5e8f\u65f6\u53ef\u80fd\u8981\u8dd1\u6ee1\u6240\u6709\u5faa\u73af\u3002\u56e0\u6b64\u9700\u8981\u5206\u4e09\u79cd\u60c5\u51b5\u8ba8\u8bba\u3002<\/p>\n<h4>15.1 \u5f62\u5f0f\u5316\u5b9a\u4e49&#xff08;\u5b9a\u4e494&#xff09;<\/h4>\n<p>\u8bbe\u4e00\u4e2a\u7b97\u6cd5\u7684\u8f93\u5165\u89c4\u6a21\u4e3a n&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          D<\/p>\n<p>          n<\/p>\n<p>        D_n<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u6240\u6709\u8f93\u5165\u7684\u96c6\u5408&#xff0c;\u4efb\u4e00\u8f93\u5165 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         I<\/p>\n<p>         \u2208<\/p>\n<p>          D<\/p>\n<p>          n<\/p>\n<p>        I \\\\in D_n<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7224em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0278em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>        P(I)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u662f I \u51fa\u73b0\u7684\u6982\u7387&#xff0c;\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         1<\/p>\n<p>        \\\\sum P(I) &#061; 1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>&#xff1b;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>        T(I)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u662f\u7b97\u6cd5\u5728\u8f93\u5165 I \u4e0b\u6240\u6267\u884c\u7684\u57fa\u672c\u8bed\u53e5\u6b21\u6570&#xff0c;\u5219\u8be5\u7b97\u6cd5\u7684\u5e73\u5747\u6267\u884c\u65f6\u95f4\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           I<\/p>\n<p>           \u2208<\/p>\n<p>            D<\/p>\n<p>            n<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         \u22c5<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>        A(n)&#061;\\\\sum_{I \\\\in D_n} P(I) \\\\cdot T(I)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4444em;vertical-align: -1.3944em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1645em\"><span class=\"\" style=\"top: -2.357em;margin-left: -0.0278em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7b97\u6cd5\u7684\u5e73\u5747\u60c5\u51b5\u6307\u5404\u79cd\u7279\u5b9a\u8f93\u5165\u4e0b\u7684\u57fa\u672c\u8bed\u53e5\u6267\u884c\u6b21\u6570\u7684\u5e26\u6743\u5e73\u5747\u503c&#xff08;\u6743\u5c31\u662f\u5404\u8f93\u5165\u51fa\u73b0\u7684\u6982\u7387&#xff09;\u3002<\/p>\n<p>\u6700\u597d\u60c5\u51b5&#xff08;Best Case&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         G<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>           M<\/p>\n<p>           I<\/p>\n<p>           N<\/p>\n<p>           I<\/p>\n<p>           \u2208<\/p>\n<p>            D<\/p>\n<p>            n<\/p>\n<p>         {<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         }<\/p>\n<p>        G(n)&#061;\\\\mathrm{MIN}_{I \\\\in D_n}\\\\{T(I)\\\\}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0001em;vertical-align: -0.2501em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathrm\">MIN<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1645em\"><span class=\"\" style=\"top: -2.357em;margin-left: -0.0278em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2501em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6307\u7b97\u6cd5\u5728\u6240\u6709\u8f93\u5165 I \u4e0b\u6240\u6267\u884c\u57fa\u672c\u8bed\u53e5\u7684\u6700\u5c11\u6b21\u6570\u3002<\/p>\n<p>\u6700\u574f\u60c5\u51b5&#xff08;Worst Case&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         W<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>           M<\/p>\n<p>           A<\/p>\n<p>           X<\/p>\n<p>           I<\/p>\n<p>           \u2208<\/p>\n<p>            D<\/p>\n<p>            n<\/p>\n<p>         {<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         }<\/p>\n<p>        W(n)&#061;\\\\mathrm{MAX}_{I \\\\in D_n}\\\\{T(I)\\\\}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">W<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0001em;vertical-align: -0.2501em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathrm\">MAX<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3283em\"><span class=\"\" style=\"top: -2.55em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1645em\"><span class=\"\" style=\"top: -2.357em;margin-left: -0.0278em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2501em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">{<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)}<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6307\u7b97\u6cd5\u5728\u6240\u6709\u8f93\u5165 I \u4e0b\u6240\u6267\u884c\u57fa\u672c\u8bed\u53e5\u7684\u6700\u5927\u6b21\u6570\u3002<\/p>\n<h4>15.2 \u4e09\u79cd\u60c5\u51b5\u7684\u610f\u4e49<\/h4>\n<table>\n<tr>\u60c5\u51b5\u8bb0\u53f7\u6570\u5b66\u5b9a\u4e49\u5de5\u7a0b\u610f\u4e49<\/tr>\n<tbody>\n<tr>\n<td>\u6700\u597d\u60c5\u51b5<\/td>\n<td>G(n)<\/td>\n<td>\u6240\u6709\u8f93\u5165\u4e2d T(I) \u7684\u6700\u5c0f\u503c<\/td>\n<td>\u4e50\u89c2\u4f30\u8ba1&#xff1b;\u901a\u5e38\u53c2\u8003\u4ef7\u503c\u4e0d\u5927&#xff08;\u597d\u8fd0\u6c14\u53ef\u9047\u4e0d\u53ef\u6c42&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u6700\u574f\u60c5\u51b5<\/td>\n<td>W(n)<\/td>\n<td>\u6240\u6709\u8f93\u5165\u4e2d T(I) \u7684\u6700\u5927\u503c<\/td>\n<td>\u6027\u80fd\u4fdd\u8bc1&#xff1a;\u7cfb\u7edf\u7edd\u4e0d\u4f1a\u518d\u6bd4\u8fd9\u66f4\u6162&#xff1b;\u5b9e\u65f6\u7cfb\u7edf\u3001\u5b89\u5168\u654f\u611f\u573a\u666f\u5fc5\u987b\u770b\u6700\u574f\u60c5\u51b5<\/td>\n<\/tr>\n<tr>\n<td>\u5e73\u5747\u60c5\u51b5<\/td>\n<td>A(n)<\/td>\n<td>T(I) \u6309\u6982\u7387\u7684\u5e26\u6743\u5e73\u5747<\/td>\n<td>\u6700\u8d34\u8fd1\u5b9e\u9645\u8fd0\u884c\u8868\u73b0&#xff1b;\u4f46\u6982\u7387\u5206\u5e03\u5f80\u5f80\u96be\u4ee5\u786e\u5b9a&#xff0c;\u8ba1\u7b97\u4e5f\u66f4\u590d\u6742<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u26a0\ufe0f \u8003\u8bd5\u5e38\u8003\u6982\u5ff5\u9898&#xff1a;\u4e09\u79cd\u60c5\u51b5\u662f\u5bf9\u540c\u4e00\u7b97\u6cd5\u5728\u4e0d\u540c\u8f93\u5165\u4e0b\u7684\u5206\u6790&#xff0c;\u800c\u4e0d\u662f&#034;\u4e09\u79cd\u4e0d\u540c\u7684\u7b97\u6cd5&#034;&#xff1b;\u6700\u574f\u60c5\u51b5\u662f\u4e0a\u754c\u4fdd\u8bc1&#xff0c;\u5e73\u5747\u60c5\u51b5\u9700\u8981\u5df2\u77e5\u8f93\u5165\u7684\u6982\u7387\u5206\u5e03\u3002<\/p>\n<hr \/>\n<h3>\u5341\u516d\u3001\u3010\u4f8b1-4\u3011\u987a\u5e8f\u67e5\u627e\u7684\u4e09\u79cd\u60c5\u51b5\u5206\u6790&#xff08;\u91cd\u70b9&#043;\u96be\u70b9&#xff09;<\/h3>\n<h4>16.1 \u9898\u76ee<\/h4>\n<p>\u91c7\u7528\u987a\u5e8f\u67e5\u627e\u65b9\u6cd5&#xff0c;\u5728\u957f\u5ea6\u4e3a n \u7684\u4e00\u7ef4\u5b9e\u578b\u6570\u7ec4 a[0..n-1] \u4e2d\u67e5\u627e\u503c\u4e3a x \u7684\u5143\u7d20&#xff1a;\u4ece\u6570\u7ec4\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\u5f00\u59cb&#xff0c;\u9010\u4e2a\u4e0e\u88ab\u67e5\u503c x \u8fdb\u884c\u6bd4\u8f83&#xff0c;\u627e\u5230\u540e\u8fd4\u56de1&#xff0c;\u5426\u5219\u8fd4\u56de0\u3002\u5bf9\u5e94\u7b97\u6cd5\u5982\u4e0b&#xff1a;<\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">Find<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">double<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">double<\/span> x<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;&#061;<\/span> x<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">break<\/span><span class=\"token punctuation\">;<\/span>    <span class=\"token comment\">\/\/ \u57fa\u672c\u8bed\u53e5<\/span><br \/>\n        i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">return<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span>         <span class=\"token comment\">\/\/ \u67e5\u627e\u6210\u529f<\/span><br \/>\n    <span class=\"token keyword\">else<\/span> <span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span>               <span class=\"token comment\">\/\/ \u67e5\u627e\u5931\u8d25<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u56de\u7b54\u4ee5\u4e0b\u95ee\u9898&#xff1a;<\/p>\n<p>(1) \u5206\u6790\u8be5\u7b97\u6cd5\u5728\u7b49\u6982\u7387\u60c5\u51b5\u4e0b\u6210\u529f\u67e5\u627e\u5230\u503c\u4e3a x \u7684\u5143\u7d20\u7684\u6700\u597d\u3001\u6700\u574f\u548c\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n<p>(2) \u5047\u8bbe\u88ab\u67e5\u503c x \u5728\u6570\u7ec4 a \u4e2d\u7684\u6982\u7387\u662f q&#xff08;\u5373\u67e5\u627e\u6210\u529f\u7684\u6982\u7387\u4e3a q&#xff0c;\u5931\u8d25\u6982\u7387\u4e3a 1\u2212q&#xff09;&#xff0c;\u6c42\u7b97\u6cd5\u7684&#xff08;\u5e73\u5747&#xff09;\u65f6\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n<h4>16.2 \u7b2c(1)\u95ee\u89e3\u7b54<\/h4>\n<p>\u7b97\u6cd5 while \u5faa\u73af\u4e2d\u7684 if \u8bed\u53e5\u662f\u57fa\u672c\u8bed\u53e5&#xff08;\u6bd4\u8f83 a[i]&#061;&#061;x \u7684\u6b21\u6570\u51b3\u5b9a\u8fd0\u884c\u65f6\u95f4&#xff09;\u3002\u6570\u7ec4 a \u4e2d\u6709 n \u4e2a\u5143\u7d20&#xff1a;<\/p>\n<p>\u6700\u597d\u60c5\u51b5&#xff1a;\u5f53\u7b2c\u4e00\u4e2a\u5143\u7d20 a[0] \u5c31\u7b49\u4e8e x \u65f6&#xff0c;\u57fa\u672c\u8bed\u53e5\u4ec5\u6267\u884c 1 \u6b21&#xff0c;\u6b64\u65f6\u5448\u73b0\u6700\u597d\u7684\u60c5\u51b5&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         G<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>        G(n) &#061; O(1)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">G<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6700\u574f\u60c5\u51b5&#xff1a;\u5f53 a \u4e2d\u6700\u540e\u4e00\u4e2a\u5143\u7d20 a[n-1] \u624d\u7b49\u4e8e x \u65f6&#xff0c;\u57fa\u672c\u8bed\u53e5\u6267\u884c n \u6b21&#xff0c;\u6b64\u65f6\u5448\u73b0\u6700\u574f\u7684\u60c5\u51b5&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         W<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        W(n) &#061; O(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">W<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5e73\u5747\u60c5\u51b5&#xff08;\u6210\u529f\u67e5\u627e\u3001\u7b49\u6982\u7387&#xff09;&#xff1a;\u5047\u8bbe\u67e5\u627e\u6bcf\u4e2a\u5143\u7d20\u7684\u6982\u7387\u76f8\u540c&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>        (<\/p>\n<p>        a<\/p>\n<p>        [<\/p>\n<p>        i<\/p>\n<p>        ]<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>          1<\/p>\n<p>          n<\/p>\n<p>        \u00a0<\/p>\n<p>        (<\/p>\n<p>        0<\/p>\n<p>        \u2264<\/p>\n<p>        i<\/p>\n<p>        \u2264<\/p>\n<p>        n<\/p>\n<p>        \u2212<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>       P(a[i]) &#061; \\\\dfrac{1}{n} \\\\ (0 \\\\le i \\\\le n-1)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mclose\">])<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">0<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7955em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u800c\u6210\u529f\u627e\u5230 a[i] \u5143\u7d20\u65f6\u57fa\u672c\u8bed\u53e5\u6b63\u597d\u6267\u884c i&#043;1 \u6b21&#xff08;\u6bd4\u8f83\u4e86 a[0]\u3001a[1]\u3001\u2026\u3001a[i] \u5171 i&#043;1 \u6b21&#xff09;&#xff0c;\u6240\u4ee5&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          1<\/p>\n<p>          n<\/p>\n<p>         (<\/p>\n<p>         i<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          1<\/p>\n<p>          n<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>         (<\/p>\n<p>         i<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          1<\/p>\n<p>          n<\/p>\n<p>         \u22c5<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         A(n)&#061;\\\\sum_{i&#061;0}^{n-1}\\\\frac{1}{n}(i&#043;1)&#061;\\\\frac{1}{n}\\\\sum_{i&#061;0}^{n-1}(i&#043;1)&#061;\\\\frac{1}{n}\\\\cdot\\\\frac{n(n&#043;1)}{2}&#061;\\\\frac{n&#043;1}{2}&#061;O(n) <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0788em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8011em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0788em;vertical-align: -1.2777em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8011em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7ed3\u8bba&#xff1a;\u7b49\u6982\u7387\u6210\u529f\u67e5\u627e\u4e0b&#xff0c;\u987a\u5e8f\u67e5\u627e\u7684\u6700\u597d\u60c5\u51b5\u4e3a O(1)&#xff0c;\u6700\u574f\u60c5\u51b5\u4e3a O(n)&#xff0c;\u5e73\u5747\u60c5\u51b5\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>          2<\/p>\n<p>       \\\\dfrac{n&#043;1}{2}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373 O(n)\u3002\u5e73\u5747\u6bd4\u8f83\u6b21\u6570\u7ea6\u4e3a\u6700\u574f\u60c5\u51b5\u7684\u4e00\u534a\u3002<\/p>\n<h4>16.3 \u7b2c(2)\u95ee\u89e3\u7b54&#xff08;\u8003\u8651\u67e5\u627e\u5931\u8d25&#xff09;<\/h4>\n<p>\u5f53\u88ab\u67e5\u503c x \u5728\u6570\u7ec4 a \u4e2d\u7684\u6982\u7387\u4e3a q \u65f6&#xff0c;\u7b97\u6cd5\u6267\u884c\u5171\u6709 n&#043;1 \u79cd\u60c5\u51b5&#xff1a;n \u79cd\u6210\u529f\u67e5\u627e&#xff08;x \u7b49\u4e8e a[0]\u2026a[n-1] \u4e2d\u67d0\u4e00\u4e2a&#xff09;\u548c 1 \u79cd\u4e0d\u6210\u529f\u67e5\u627e&#xff08;x \u4e0d\u5728\u6570\u7ec4\u4e2d&#xff09;\u3002<\/p>\n<ul>\n<li>\u6210\u529f\u67e5\u627e&#xff1a;\u5047\u8bbe\u7b49\u6982\u7387&#xff0c;\u5219\u5143\u7d20 a[i] \u88ab\u67e5\u627e\u5230\u7684\u6982\u7387\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         a<\/p>\n<p>         [<\/p>\n<p>         i<\/p>\n<p>         ]<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>           q<\/p>\n<p>           n<\/p>\n<p>        P(a[i]) &#061; \\\\dfrac{q}{n}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mopen\">[<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mclose\">])<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.7936em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1076em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;\u603b\u6210\u529f\u6982\u7387 q \u5747\u644a\u5230 n \u4e2a\u5143\u7d20\u4e0a&#xff09;&#xff0c;\u6210\u529f\u627e\u5230 a[i] \u65f6\u57fa\u672c\u8bed\u53e5\u6b63\u597d\u6267\u884c i&#043;1 \u6b21&#xff1b;<\/li>\n<li>\u4e0d\u6210\u529f\u67e5\u627e&#xff1a;\u5176\u6982\u7387\u4e3a 1\u2212q&#xff0c;\u4e0d\u6210\u529f\u67e5\u627e\u65f6\u5faa\u73af\u8d70\u5b8c\u5168\u7a0b&#xff0c;\u57fa\u672c\u8bed\u53e5\u6b63\u597d\u6267\u884c n \u6b21&#xff08;i \u4ece 0 \u6bd4\u8f83\u5230 n\u22121 \u90fd\u4e0d\u5339\u914d&#xff0c;i&lt;n \u4e0d\u6210\u7acb\u9000\u51fa&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u6240\u4ee5&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           I<\/p>\n<p>           \u2208<\/p>\n<p>            D<\/p>\n<p>            n<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         \u22c5<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           0<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          q<\/p>\n<p>          n<\/p>\n<p>         (<\/p>\n<p>         i<\/p>\n<p>         &#043;<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         \u2212<\/p>\n<p>         q<\/p>\n<p>         )<br \/>\n        \u2009<\/p>\n<p>         n<\/p>\n<p>         &#061;<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           q<\/p>\n<p>          2<\/p>\n<p>         &#043;<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         \u2212<\/p>\n<p>         q<\/p>\n<p>         )<\/p>\n<p>         n<\/p>\n<p>         A(n)&#061;\\\\sum_{I \\\\in D_n} P(I)\\\\cdot T(I)&#061;\\\\sum_{i&#061;0}^{n-1}\\\\frac{q}{n}(i&#043;1)&#043;(1-q)\\\\,n&#061;\\\\frac{(n&#043;1)q}{2}&#043;(1-q)n <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.4444em;vertical-align: -1.3944em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.05em\"><span class=\"\" style=\"top: -1.8557em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mrel mtight\">\u2208<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0278em\">D<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1645em\"><span class=\"\" style=\"top: -2.357em;margin-left: -0.0278em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3.0788em;vertical-align: -1.2777em\"><\/span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.8011em\"><span class=\"\" style=\"top: -1.8723em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.05em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"\"><span class=\"mop op-symbol large-op\">\u2211<\/span><\/span><\/span><span class=\"\" style=\"top: -4.3em;margin-left: 0em\"><span class=\"pstrut\" style=\"height: 3.05em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.2777em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1076em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7279\u4f8b&#xff1a;\u5982\u679c\u5df2\u77e5\u9700\u8981\u67e5\u627e\u7684 x \u6709\u4e00\u534a\u7684\u673a\u4f1a\u5728\u6570\u7ec4\u4e2d&#xff0c;\u5373 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        q<\/p>\n<p>        &#061;<\/p>\n<p>        1<\/p>\n<p>        \/<\/p>\n<p>        2<\/p>\n<p>       q &#061; 1\/2<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">q<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1\/2<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>          4<\/p>\n<p>         &#043;<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         \u2248<\/p>\n<p>           3<\/p>\n<p>           n<\/p>\n<p>          4<\/p>\n<p>         A(n)&#061;\\\\frac{n&#043;1}{4}&#043;\\\\frac{n}{2}\\\\approx \\\\frac{3n}{4} <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.7936em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1076em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7ed3\u679c\u89e3\u8bfb&#xff1a;<\/p>\n<ul>\n<li>\u5f53 q&#061;1&#xff08;\u4e00\u5b9a\u67e5\u627e\u6210\u529f&#xff09;\u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>            n<\/p>\n<p>            &#043;<\/p>\n<p>            1<\/p>\n<p>           2<\/p>\n<p>         \u2248<\/p>\n<p>           n<\/p>\n<p>           2<\/p>\n<p>        A(n)&#061;\\\\dfrac{n&#043;1}{2}\\\\approx\\\\dfrac{n}{2}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.7936em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1076em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li>\u5f53 q&#061;1\/2 \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         \u2248<\/p>\n<p>            3<\/p>\n<p>            n<\/p>\n<p>           4<\/p>\n<p>        A(n)\\\\approx\\\\dfrac{3n}{4}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li>\u5f53 q&#061;0&#xff08;\u4e00\u5b9a\u5931\u8d25&#xff09;\u65f6&#xff0c;A(n)&#061;n\u3002<\/li>\n<\/ul>\n<p>\u6210\u529f\u6982\u7387\u8d8a\u4f4e&#xff0c;\u5e73\u5747\u6bd4\u8f83\u6b21\u6570\u8d8a\u591a\u2014\u2014\u56e0\u4e3a\u5931\u8d25\u67e5\u627e\u603b\u8981\u628a\u6574\u4e2a\u6570\u7ec4\u626b\u5b8c&#xff08;n \u6b21\u6bd4\u8f83&#xff09;&#xff0c;\u662f\u6700&#034;\u6602\u8d35&#034;\u7684\u60c5\u5f62\u3002\u8fd9\u4e2a\u7ed3\u8bba\u975e\u5e38\u7b26\u5408\u76f4\u89c9&#xff0c;\u4e5f\u63d0\u9192\u6211\u4eec&#xff1a;\u5206\u6790\u5e73\u5747\u60c5\u51b5\u65f6\u5fc5\u987b\u628a\u5931\u8d25\u8def\u5f84\u8ba1\u5165\u3002<\/p>\n<h4>16.4 \u672c\u4f8b\u65b9\u6cd5\u8bba\u603b\u7ed3<\/h4>\n<p>\u5206\u6790&#034;\u4e09\u79cd\u60c5\u51b5&#034;\u9898\u76ee\u7684\u6807\u51c6\u5957\u8def&#xff1a;<\/p>\n<li>\u786e\u5b9a\u57fa\u672c\u8bed\u53e5&#xff08;\u901a\u5e38\u662f\u6700\u5185\u5c42\u5faa\u73af\u4e2d\u7684\u6bd4\u8f83\/\u8d4b\u503c\u8bed\u53e5&#xff09;&#xff1b;<\/li>\n<li>\u6700\u597d\u60c5\u51b5&#xff1a;\u627e\u8ba9\u57fa\u672c\u8bed\u53e5\u6267\u884c\u6b21\u6570\u6700\u5c11\u7684\u8f93\u5165&#xff08;\u5982\u9996\u5143\u7d20\u5373\u547d\u4e2d&#xff09;&#xff1b;<\/li>\n<li>\u6700\u574f\u60c5\u51b5&#xff1a;\u627e\u8ba9\u57fa\u672c\u8bed\u53e5\u6267\u884c\u6b21\u6570\u6700\u591a\u7684\u8f93\u5165&#xff08;\u5982\u672b\u5143\u7d20\u547d\u4e2d\u6216\u6839\u672c\u4e0d\u547d\u4e2d&#xff09;&#xff1b;<\/li>\n<li>\u5e73\u5747\u60c5\u51b5&#xff1a;\u5217\u51fa\u6240\u6709\u8f93\u5165\u60c5\u5f62\u3001\u5404\u81ea\u6982\u7387 P(I) \u548c\u6267\u884c\u6b21\u6570 T(I)&#xff0c;\u4ee3\u5165 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2211<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>        A(n)&#061;\\\\sum P(I)T(I)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6c42\u548c&#xff0c;\u7528\u7b49\u5dee\/\u7b49\u6bd4\u6570\u5217\u516c\u5f0f\u5316\u7b80&#xff1b;<\/li>\n<li>\u6ce8\u610f\u533a\u5206&#034;\u4ec5\u6210\u529f\u67e5\u627e\u7684\u5e73\u5747&#034;\u4e0e&#034;\u6210\u529f&#043;\u5931\u8d25\u6df7\u5408\u7684\u5e73\u5747&#034;\u2014\u2014\u540e\u8005\u8981\u628a\u5931\u8d25\u60c5\u5f62\u7684\u6982\u7387 (1\u2212q) \u548c\u6b21\u6570 n \u52a0\u8fdb\u53bb\u3002<\/li>\n<hr \/>\n<h3>\u5341\u4e03\u3001\u975e\u9012\u5f52\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790<\/h3>\n<h4>17.1 \u65b9\u6cd5\u6982\u8ff0<\/h4>\n<p>\u975e\u9012\u5f52\u7b97\u6cd5&#xff08;\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790&#xff09;\u7684\u5173\u952e\u662f\u6c42\u51fa\u4ee3\u8868\u7b97\u6cd5\u6267\u884c\u65f6\u95f4\u7684\u8868\u8fbe\u5f0f f(n)\u3002<\/p>\n<p>\u4e00\u822c\u6b65\u9aa4&#xff1a;<\/p>\n<li>\u627e\u51fa\u57fa\u672c\u8bed\u53e5&#xff08;\u6700\u5185\u5c42\u5faa\u73af\u4f53\u5185\u7684\u8bed\u53e5&#xff09;&#xff1b;<\/li>\n<li>\u8bbe\u57fa\u672c\u8bed\u53e5\u6267\u884c\u6b21\u6570\u4e3a m&#xff0c;\u5efa\u7acb m \u4e0e\u95ee\u9898\u89c4\u6a21 n \u7684\u5173\u7cfb\u5f0f&#xff08;\u7b49\u5f0f\u6216\u4e0d\u7b49\u5f0f&#xff09;&#xff1b;<\/li>\n<li>\u89e3\u51fa m \u5173\u4e8e n \u7684\u8868\u8fbe\u5f0f f(n)&#xff1b;<\/li>\n<li>\u7528\u6e10\u8fdb\u7b26\u53f7\u53d6\u5176\u9636\u3002<\/li>\n<p>\u5bf9\u4e8e\u5faa\u73af\u53d8\u91cf\u975e\u7ebf\u6027\u53d8\u5316&#xff08;\u5982 i &#043;&#061; 2\u3001i *&#061; 2\u3001i &#061; i*i&#xff09;\u7684\u5faa\u73af&#xff0c;\u8fd9\u662f\u6700\u5e38\u8003\u7684\u9898\u578b&#xff0c;\u4e0b\u9762\u7528\u4f8b\u9898\u793a\u8303\u3002<\/p>\n<h4>17.2 \u3010\u4f8b1-5\u3011<\/h4>\n<p>\u7ed9\u51fa\u4ee5\u4e0b\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">func<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">100<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        k<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        i <span class=\"token operator\">&#043;&#061;<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u89e3&#xff1a;\u57fa\u672c\u8bed\u53e5\u662f while \u5faa\u73af\u4f53\u5185\u7684\u8bed\u53e5&#xff08;k&#043;&#043;; i&#043;&#061;2;&#xff09;\u3002\u8bbe\u5176\u6267\u884c\u6b21\u6570\u4e3a m\u3002<\/p>\n<p>\u8ffd\u8e2a\u5faa\u73af\u53d8\u91cf i \u7684\u53d8\u5316\u89c4\u5f8b&#xff1a;i \u4ece 1 \u5f00\u59cb&#xff0c;\u6bcf\u6b21\u9012\u589e2&#xff0c;\u6267\u884c m \u6b21\u540e&#xff0c;i \u7684\u53d6\u503c\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         i<\/p>\n<p>         &#061;<\/p>\n<p>         1<\/p>\n<p>         &#043;<\/p>\n<p>         2<\/p>\n<p>         m<\/p>\n<p>        i &#061; 1 &#043; 2m<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6595em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5faa\u73af\u7ee7\u7eed\u7684\u6761\u4ef6\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        i<\/p>\n<p>        \u2264<\/p>\n<p>        n<\/p>\n<p>       i \\\\le n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7955em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u56e0\u6b64\u6709&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         i<\/p>\n<p>         &#061;<\/p>\n<p>         1<\/p>\n<p>         &#043;<\/p>\n<p>         2<\/p>\n<p>         m<\/p>\n<p>         \u2264<\/p>\n<p>         n<br \/>\n        \u2005\u200a<\/p>\n<p>         \u21d2<br \/>\n        \u2005\u200a<\/p>\n<p>         m<\/p>\n<p>         \u2264<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          2<\/p>\n<p>        i &#061; 1&#043;2m \\\\le n \\\\;\\\\Rightarrow\\\\; m \\\\le \\\\frac{n-1}{2}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6595em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7804em;vertical-align: -0.136em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         m<\/p>\n<p>         \u2264<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; m \\\\le \\\\frac{n-1}{2} &#061; O(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.0074em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.3214em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8be5\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        O(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>17.3 \u53d8\u5f0f\u8bad\u7ec3&#xff08;\u9884\u4e60\u81ea\u6d4b&#xff09;<\/h4>\n<p>\u638c\u63e1\u4e86\u4f8b1-5\u7684\u65b9\u6cd5&#xff0c;\u4e0b\u9762\u51e0\u4e2a\u7ecf\u5178\u53d8\u5f0f\u5e94\u8be5\u90fd\u80fd\u79d2\u7b54&#xff1a;<\/p>\n<p>\u53d8\u5f0f1&#xff1a;i &#061; 1; while (i &lt;&#061; n) i &#061; i * 2; \u8bbe\u6267\u884c m \u6b21\u540e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        i<\/p>\n<p>        &#061;<\/p>\n<p>         2<\/p>\n<p>         m<\/p>\n<p>        \u2264<\/p>\n<p>        n<\/p>\n<p>       i &#061; 2^m \\\\le n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6595em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8004em;vertical-align: -0.136em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        m<\/p>\n<p>        \u2264<\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>       m \\\\le \\\\log_2 n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        O(\\\\log_2 n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u53d8\u5f0f2&#xff1a;i &#061; 0; while (i*i &lt;&#061; n) i&#043;&#043;; \u8bbe\u6267\u884c m \u6b21\u540e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         m<\/p>\n<p>         2<\/p>\n<p>        \u2264<\/p>\n<p>        n<\/p>\n<p>       m^2 \\\\le n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9501em;vertical-align: -0.136em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">m<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        m<\/p>\n<p>        \u2264<\/p>\n<p>         n<\/p>\n<p>       m \\\\le \\\\sqrt{n}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7719em;vertical-align: -0.136em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2264<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.04em;vertical-align: -0.2397em\"><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8003em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -2.7603em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/p>\n<p>           <\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2397em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>         )<\/p>\n<p>        O(\\\\sqrt{n})<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0503em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8003em\"><span class=\"svg-align\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\" style=\"padding-left: 0.833em\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -2.7603em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"hide-tail\" style=\"min-width: 0.853em;height: 1.08em\"><\/p>\n<p>            <\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2397em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u53d8\u5f0f3&#xff1a;i &#061; n; while (i &gt; 1) i &#061; i \/ 2; \u8bbe\u6267\u884c m \u6b21\u540e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \/<\/p>\n<p>         2<\/p>\n<p>         m<\/p>\n<p>        &gt;<\/p>\n<p>        1<\/p>\n<p>       n\/2^m &gt; 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        m<\/p>\n<p>        &lt;<\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>       m &lt; \\\\log_2 n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5782em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&lt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u65f6\u95f4\u590d\u6742\u5ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        O(\\\\log_2 n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u53d8\u5f0f4&#xff1a;\u53cc\u91cd\u5faa\u73af&#xff0c;\u5916\u5c42 for(i&#061;0;i&lt;n;i&#043;&#043;)&#xff0c;\u5185\u5c42 for(j&#061;0;j&lt;i;j&#043;&#043;)&#xff0c;\u57fa\u672c\u8bed\u53e5\u5728\u5185\u5c42\u3002 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>         \u2211<\/p>\n<p>          i<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>          n<\/p>\n<p>          \u2212<\/p>\n<p>          1<\/p>\n<p>        i<\/p>\n<p>        &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>         n<\/p>\n<p>         2<\/p>\n<p>        )<\/p>\n<p>       f(n)&#061;\\\\sum_{i&#061;0}^{n-1} i &#061; \\\\dfrac{n(n-1)}{2} &#061; O(n^2)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2537em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.954em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 2.113em;vertical-align: -0.686em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.427em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>&#x1f4a1; \u6838\u5fc3\u5fc3\u6cd5&#xff1a;\u4e0d\u7ba1\u5faa\u73af\u53d8\u91cf\u600e\u4e48\u8df3&#xff08;&#043;2\u3001\u00d72\u3001\u00f72\u3001\u5e73\u65b9&#xff09;&#xff0c;\u5957\u8def\u6c38\u8fdc\u662f\u2014\u2014\u201c\u8bbe\u6267\u884c m \u6b21&#xff0c;\u5199\u51fa\u7b2c m \u6b21\u540e\u5faa\u73af\u53d8\u91cf\u7684\u8868\u8fbe\u5f0f&#xff0c;\u4ee3\u5165\u5faa\u73af\u6761\u4ef6\u89e3\u51fa m \u5173\u4e8e n \u7684\u4e0a\u754c\u201d\u3002<\/p>\n<hr \/>\n<h3>\u5341\u516b\u3001\u9012\u5f52\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790<\/h3>\n<h4>18.1 \u65b9\u6cd5\u6982\u8ff0<\/h4>\n<p>\u9012\u5f52\u7b97\u6cd5\u662f\u91c7\u7528\u4e00\u79cd\u5206\u800c\u6cbb\u4e4b\u7684\u65b9\u6cd5&#xff0c;\u628a\u4e00\u4e2a&#034;\u5927\u95ee\u9898&#034;\u5206\u89e3\u4e3a\u82e5\u5e72\u4e2a\u76f8\u4f3c\u7684&#034;\u5c0f\u95ee\u9898&#034;\u6765\u6c42\u89e3\u3002<\/p>\n<p>\u5206\u6790\u9012\u5f52\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5173\u952e\u662f&#xff1a;\u6839\u636e\u9012\u5f52\u8fc7\u7a0b\u5efa\u7acb\u9012\u63a8\u5173\u7cfb\u5f0f&#xff0c;\u7136\u540e\u6c42\u89e3\u8fd9\u4e2a\u9012\u63a8\u5173\u7cfb\u5f0f&#xff0c;\u5f97\u5230\u4e00\u4e2a\u8868\u793a\u7b97\u6cd5\u6267\u884c\u65f6\u95f4\u7684\u8868\u8fbe\u5f0f&#xff0c;\u6700\u540e\u7528\u6e10\u8fdb\u7b26\u53f7\u6765\u8868\u793a\u8fd9\u4e2a\u8868\u8fbe\u5f0f&#xff0c;\u5373\u5f97\u5230\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n<p>\u6807\u51c6\u56db\u6b65\u6cd5&#xff1a;<\/p>\n<li>\u8bbe\u89c4\u6a21\u4e3a n \u7684\u8c03\u7528\u6240\u9700\u65f6\u95f4\u4e3a T(n)&#xff1b;<\/li>\n<li>\u6839\u636e&#034;\u9012\u5f52\u8c03\u7528\u7684\u5b50\u95ee\u9898\u89c4\u6a21\u4e0e\u4e2a\u6570 &#043; \u9664\u9012\u5f52\u5916\u7684\u5176\u4ed6\u5de5\u4f5c\u91cf&#034;\u5199\u51fa\u9012\u63a8\u5173\u7cfb\u5f0f&#xff08;\u542b\u521d\u59cb\u6761\u4ef6&#xff09;&#xff1b;<\/li>\n<li>\u7528\u8fed\u4ee3\u5c55\u5f00\u6cd5&#xff08;\u6216\u4e3b\u5b9a\u7406&#xff09;\u6c42\u89e3\u9012\u63a8\u5f0f&#xff1b;<\/li>\n<li>\u7528\u5927O\u8868\u793a\u7ed3\u679c\u3002<\/li>\n<h4>18.2 \u3010\u4f8b1-6\u3011\u5f52\u5e76\u6392\u5e8f\u7684\u9012\u63a8\u5206\u6790<\/h4>\n<p>\u9898\u76ee&#xff1a;\u6709\u4ee5\u4e0b\u9012\u5f52\u7b97\u6cd5&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">mergesort<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> j<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> m<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">!&#061;<\/span> j<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        m <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#043;<\/span> j<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token function\">mergesort<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span>        <span class=\"token comment\">\/\/ \u9012\u5f52\u6392\u5e8f\u5de6\u534a\u90e8\u5206<\/span><br \/>\n        <span class=\"token function\">mergesort<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> m <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span>    <span class=\"token comment\">\/\/ \u9012\u5f52\u6392\u5e8f\u53f3\u534a\u90e8\u5206<\/span><br \/>\n        <span class=\"token function\">merge<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">,<\/span> m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span>         <span class=\"token comment\">\/\/ \u5408\u5e76\u4e24\u4e2a\u6709\u5e8f\u5b50\u5e8f\u5217<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>mergesort() \u7528\u4e8e\u6570\u7ec4 a[0..n-1]&#xff08;\u8bbe <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        &#061;<\/p>\n<p>         2<\/p>\n<p>         k<\/p>\n<p>       n&#061;2^k<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;k \u4e3a\u6b63\u6574\u6570&#xff09;\u7684\u5f52\u5e76\u6392\u5e8f&#xff0c;\u8c03\u7528\u65b9\u5f0f\u4e3a mergesort(a, 0, n-1)&#xff1b;\u53e6\u5916 merge(a, i, j, m) \u7528\u4e8e\u4e24\u4e2a\u6709\u5e8f\u5b50\u5e8f\u5217\u7684\u6709\u5e8f\u5408\u5e76&#xff0c;\u662f\u975e\u9012\u5f52\u51fd\u6570&#xff0c;\u5b83\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       O(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff08;\u8fd9\u91cc <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        &#061;<\/p>\n<p>        j<\/p>\n<p>        \u2212<\/p>\n<p>        i<\/p>\n<p>        &#043;<\/p>\n<p>        1<\/p>\n<p>       n &#061; j-i&#043;1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.854em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0572em\">j<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7429em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373\u5f53\u524d\u5904\u7406\u7684\u533a\u95f4\u957f\u5ea6&#xff09;\u3002\u5206\u6790\u4e0a\u8ff0\u8c03\u7528\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n<p>\u89e3&#xff1a;\u8bbe\u8c03\u7528 mergesort(a, 0, n-1) \u7684\u6267\u884c\u65f6\u95f4\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        T<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       T(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u7531\u5176\u6267\u884c\u8fc7\u7a0b&#xff08;\u89c4\u6a21 n \u7684\u95ee\u9898\u5206\u89e3\u4e3a\u4e24\u4e2a\u89c4\u6a21 n\/2 \u7684\u5b50\u95ee\u9898&#xff0c;\u52a0\u4e0a\u4e00\u6b21 O(n) \u7684\u5408\u5e76&#xff09;\u5f97\u5230\u4ee5\u4e0b\u6c42\u6267\u884c\u65f6\u95f4\u7684\u9012\u5f52\u5173\u7cfb&#xff08;\u9012\u63a8\u5173\u7cfb\u5f0f&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          {<\/p>\n<p>               O<\/p>\n<p>               (<\/p>\n<p>               1<\/p>\n<p>               )<\/p>\n<p>               \u5f53\u00a0<\/p>\n<p>               n<\/p>\n<p>               &#061;<\/p>\n<p>               1<\/p>\n<p>               2<\/p>\n<p>               T<\/p>\n<p>               (<\/p>\n<p>               n<\/p>\n<p>               \/<\/p>\n<p>               2<\/p>\n<p>               )<\/p>\n<p>               &#043;<\/p>\n<p>               O<\/p>\n<p>               (<\/p>\n<p>               n<\/p>\n<p>               )<\/p>\n<p>               \u5f53\u00a0<\/p>\n<p>               n<\/p>\n<p>               &gt;<\/p>\n<p>               1<\/p>\n<p>         T(n)&#061;\\\\begin{cases} O(1) &amp; \\\\text{\u5f53 } n&#061;1 \\\\\\\\ 2T(n\/2)&#043;O(n) &amp; \\\\text{\u5f53 } n&gt;1 \\\\end{cases} <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3em;vertical-align: -1.25em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">{<\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.69em\"><span class=\"\" style=\"top: -3.69em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -2.25em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.19em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"arraycolsep\" style=\"width: 1em\"><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.69em\"><span class=\"\" style=\"top: -3.69em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u5f53<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -2.25em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u5f53<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.19em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       O(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e3a merge() \u6240\u9700\u7684\u65f6\u95f4&#xff0c;\u8bbe\u4e3a cn&#xff08;c \u4e3a\u6b63\u5e38\u91cf&#xff09;\u3002\u4e0b\u9762\u7528\u8fed\u4ee3\u5c55\u5f00\u6cd5\u6c42\u89e3&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>             T<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             )<\/p>\n<p>             &#061;<\/p>\n<p>             2<\/p>\n<p>             T<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>             2<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             c<\/p>\n<p>             n<\/p>\n<p>             &#061;<\/p>\n<p>             2<\/p>\n<p>              [<\/p>\n<p>              2<\/p>\n<p>              T<\/p>\n<p>              (<\/p>\n<p>              n<\/p>\n<p>              \/<\/p>\n<p>               2<\/p>\n<p>               2<\/p>\n<p>              )<\/p>\n<p>              &#043;<\/p>\n<p>              c<\/p>\n<p>              \u22c5<\/p>\n<p>               n<\/p>\n<p>               2<\/p>\n<p>              ]<\/p>\n<p>             &#043;<\/p>\n<p>             c<\/p>\n<p>             n<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>             T<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             2<\/p>\n<p>             c<\/p>\n<p>             n<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>              [<\/p>\n<p>              2<\/p>\n<p>              T<\/p>\n<p>              (<\/p>\n<p>              n<\/p>\n<p>              \/<\/p>\n<p>               2<\/p>\n<p>               3<\/p>\n<p>              )<\/p>\n<p>              &#043;<\/p>\n<p>              c<\/p>\n<p>              \u22c5<\/p>\n<p>               n<\/p>\n<p>                2<\/p>\n<p>                2<\/p>\n<p>              ]<\/p>\n<p>             &#043;<\/p>\n<p>             2<\/p>\n<p>             c<\/p>\n<p>             n<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              3<\/p>\n<p>             T<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>              2<\/p>\n<p>              3<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             3<\/p>\n<p>             c<\/p>\n<p>             n<\/p>\n<p>             &#061;<\/p>\n<p>             \u22ef<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              k<\/p>\n<p>             T<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>              2<\/p>\n<p>              k<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             k<\/p>\n<p>             \u22c5<\/p>\n<p>             c<\/p>\n<p>             n<\/p>\n<p>         \\\\begin{align*} T(n) &amp;&#061; 2T(n\/2)&#043;cn \\\\\\\\ &amp;&#061; 2\\\\left[2T(n\/2^2)&#043;c\\\\cdot\\\\frac{n}{2}\\\\right]&#043;cn &#061; 2^2T(n\/2^2)&#043;2cn \\\\\\\\ &amp;&#061; 2^2\\\\left[2T(n\/2^3)&#043;c\\\\cdot\\\\frac{n}{2^2}\\\\right]&#043;2cn &#061; 2^3T(n\/2^3)&#043;3cn \\\\\\\\ &amp;&#061; \\\\cdots \\\\\\\\ &amp;&#061; 2^k T(n\/2^k)&#043;k\\\\cdot cn \\\\end{align*} <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 8.8311em;vertical-align: -4.1656em\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 4.6656em\"><span class=\"\" style=\"top: -6.9756em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -5.1656em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><\/span><\/span><span class=\"\" style=\"top: -3.0296em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><\/span><\/span><span class=\"\" style=\"top: -1.2036em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><\/span><\/span><span class=\"\" style=\"top: 0.3556em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 4.1656em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 4.6656em\"><span class=\"\" style=\"top: -6.9756em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -5.1656em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">[<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1076em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">]<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -3.0296em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">[<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1076em\"><span class=\"\" style=\"top: -2.314em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7401em\"><span class=\"\" style=\"top: -2.989em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.677em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.686em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size2\">]<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><span class=\"\" style=\"top: -1.2036em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"minner\">\u22ef<\/span><\/span><\/span><span class=\"\" style=\"top: 0.3556em\"><span class=\"pstrut\" style=\"height: 3.15em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 4.1656em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u8fd9\u91cc\u5047\u8bbe <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        &#061;<\/p>\n<p>         2<\/p>\n<p>         k<\/p>\n<p>       n&#061;2^k<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        k<\/p>\n<p>        &#061;<\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>       k&#061;\\\\log_2 n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>\u3002\u5f53\u5c55\u5f00\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        n<\/p>\n<p>        \/<\/p>\n<p>         2<\/p>\n<p>         k<\/p>\n<p>        &#061;<\/p>\n<p>        1<\/p>\n<p>       n\/2^k &#061; 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u65f6&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        T<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>       T(1)&#061;O(1)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u4ee3\u5165\u5f97&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          2<\/p>\n<p>          k<\/p>\n<p>         \u22c5<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         c<\/p>\n<p>         n<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>         &#061;<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         c<\/p>\n<p>         n<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         T(n) &#061; 2^k \\\\cdot O(1) &#043; cn\\\\log_2 n &#061; n &#043; cn\\\\log_2 n &#061; O(n\\\\log_2 n) <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8991em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7ed3\u8bba&#xff1a;\u5f52\u5e76\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>             log<\/p>\n<p>             \u2061<\/p>\n<p>            2<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>        \\\\boldsymbol{O(n\\\\log_2 n)}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.0319em\">O<\/span><span class=\"mopen mathbf\">(<\/span><span class=\"mord boldsymbol\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\"><span class=\"mathbf\">l<\/span><span class=\"mathbf\">o<\/span><span class=\"mathbf\" style=\"margin-right: 0.016em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathbf mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord boldsymbol\">n<\/span><span class=\"mclose mathbf\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>18.3 \u5c55\u5f00\u8fc7\u7a0b\u7684\u89c4\u5f8b\u89c2\u5bdf<\/h4>\n<p>\u8fed\u4ee3\u5c55\u5f00\u4e2d\u6bcf\u4e00\u884c\u7684\u7ed3\u6784\u662f&#034;\u9012\u5f52\u9879 &#043; \u7d2f\u8ba1\u5de5\u4f5c\u91cf&#034;&#xff1a;<\/p>\n<table>\n<tr>\u5c55\u5f00\u6b21\u6570\u9012\u5f52\u9879\u7d2f\u8ba1\u975e\u9012\u5f52\u5de5\u4f5c\u91cf<\/tr>\n<tbody>\n<tr>\n<td>\u7b2c1\u6b21<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           2<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>           2<\/p>\n<p>           )<\/p>\n<p>          2T(n\/2)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           c<\/p>\n<p>           n<\/p>\n<p>          cn<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u7b2c2\u6b21<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>            2<\/p>\n<p>            2<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>            2<\/p>\n<p>            2<\/p>\n<p>           )<\/p>\n<p>          2^2T(n\/2^2)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           2<\/p>\n<p>           c<\/p>\n<p>           n<\/p>\n<p>          2cn<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u7b2c3\u6b21<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>            2<\/p>\n<p>            3<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>            2<\/p>\n<p>            3<\/p>\n<p>           )<\/p>\n<p>          2^3T(n\/2^3)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           3<\/p>\n<p>           c<\/p>\n<p>           n<\/p>\n<p>          3cn<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">3<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<tr>\n<td>\u7b2ck\u6b21<\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>            2<\/p>\n<p>            k<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>            2<\/p>\n<p>            k<\/p>\n<p>           )<\/p>\n<p>          2^kT(n\/2^k)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           k<\/p>\n<p>           c<\/p>\n<p>           n<\/p>\n<p>          kcn<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u89c4\u5f8b&#xff1a;\u6bcf\u5c55\u5f00\u4e00\u5c42&#xff0c;\u9012\u5f52\u9879\u7cfb\u6570\u7ffb\u500d\u3001\u89c4\u6a21\u51cf\u534a&#xff0c;\u5de5\u4f5c\u91cf\u7d2f\u52a0\u4e00\u4efd cn\u3002\u5c55\u5f00 k \u5c42\u540e\u89e6\u5e95&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        T<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>       T(1)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u6b64\u65f6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         2<\/p>\n<p>         k<\/p>\n<p>        &#061;<\/p>\n<p>        n<\/p>\n<p>       2^k &#061; n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u603b\u5de5\u4f5c\u91cf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        &#061;<\/p>\n<p>        n<\/p>\n<p>        \u22c5<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>        &#043;<\/p>\n<p>        k<\/p>\n<p>        c<\/p>\n<p>        n<\/p>\n<p>        &#061;<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        c<\/p>\n<p>        n<\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>       &#061; n\\\\cdot O(1) &#043; kcn &#061; n &#043; cn\\\\log_2 n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.3669em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u4e5f\u53ef\u4ee5\u7528\u9012\u5f52\u6811\u76f4\u89c2\u7406\u89e3&#xff1a;\u6811\u5171 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        1<\/p>\n<p>       \\\\log_2 n &#043; 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u5c42&#xff0c;\u6bcf\u4e00\u5c42\u6240\u6709\u8282\u70b9\u7684\u5408\u5e76\u5de5\u4f5c\u91cf\u4e4b\u548c\u90fd\u662f cn&#xff0c;\u6545\u603b\u5de5\u4f5c\u91cf\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        c<\/p>\n<p>        n<\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>        &#043;<\/p>\n<p>        n<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>          log<\/p>\n<p>          \u2061<\/p>\n<p>         2<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       cn\\\\log_2 n &#043; n &#061; O(n\\\\log_2 n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>18.4 \u62d3\u5c55&#xff1a;\u4e3b\u5b9a\u7406&#xff08;\u9884\u4e60\u52a0\u5206\u9879&#xff09;<\/h4>\n<p>\u5bf9\u4e8e\u5f62\u5982 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        T<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        a<\/p>\n<p>        T<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        \/<\/p>\n<p>        b<\/p>\n<p>        )<\/p>\n<p>        &#043;<\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       T(n) &#061; aT(n\/b) &#043; f(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u7684\u9012\u63a8\u5f0f&#xff0c;\u4e3b\u5b9a\u7406&#xff08;Master Theorem&#xff09; \u53ef\u4ee5\u76f4\u63a5\u7ed9\u51fa\u7b54\u6848&#xff1a;<\/p>\n<p>\u6bd4\u8f83 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       f(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e0e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         n<\/p>\n<p>            log<\/p>\n<p>            \u2061<\/p>\n<p>           b<\/p>\n<p>          a<\/p>\n<p>       n^{\\\\log_b a}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2302em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u9636&#xff1a;<\/p>\n<li>\u82e5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>             log<\/p>\n<p>             \u2061<\/p>\n<p>            b<\/p>\n<p>           a<\/p>\n<p>           \u2212<\/p>\n<p>           \u03b5<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; O(n^{\\\\log_b a &#8211; \\\\varepsilon})<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2302em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">\u03b5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff08;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03b5<\/p>\n<p>         &gt;<\/p>\n<p>         0<\/p>\n<p>        \\\\varepsilon&gt;0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5782em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathnormal\">\u03b5<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>&#xff09;&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>             log<\/p>\n<p>             \u2061<\/p>\n<p>            b<\/p>\n<p>           a<\/p>\n<p>         )<\/p>\n<p>        T(n)&#061;\\\\Theta(n^{\\\\log_b a})<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2302em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li>\u82e5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>             log<\/p>\n<p>             \u2061<\/p>\n<p>            b<\/p>\n<p>           a<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          k<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Theta(n^{\\\\log_b a}\\\\log^k n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1834em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2302em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9334em\"><span class=\"\" style=\"top: -3.1473em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>             log<\/p>\n<p>             \u2061<\/p>\n<p>            b<\/p>\n<p>           a<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           k<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        T(n)&#061;\\\\Theta(n^{\\\\log_b a}\\\\log^{k&#043;1} n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1834em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2302em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9334em\"><span class=\"\" style=\"top: -3.1473em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li>\u82e5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>             log<\/p>\n<p>             \u2061<\/p>\n<p>            b<\/p>\n<p>           a<\/p>\n<p>           &#043;<\/p>\n<p>           \u03b5<\/p>\n<p>         )<\/p>\n<p>        f(n) &#061; \\\\Omega(n^{\\\\log_b a &#043; \\\\varepsilon})<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2302em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">b<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mathnormal mtight\">a<\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mathnormal mtight\">\u03b5<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e14\u6ee1\u8db3\u6b63\u5219\u6761\u4ef6&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        T(n)&#061;\\\\Theta(f(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span>\u3002<\/li>\n<p>\u5bf9\u5f52\u5e76\u6392\u5e8f&#xff1a;a&#061;2, b&#061;2, f(n)&#061;cn\u3002<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         n<\/p>\n<p>            log<\/p>\n<p>            \u2061<\/p>\n<p>           2<\/p>\n<p>          2<\/p>\n<p>        &#061;<\/p>\n<p>         n<\/p>\n<p>         1<\/p>\n<p>        &#061;<\/p>\n<p>        n<\/p>\n<p>       n^{\\\\log_2 2}&#061;n^1&#061;n<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop mtight\"><span class=\"mtight\">l<\/span><span class=\"mtight\">o<\/span><span class=\"mtight\" style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\" style=\"top: -2.2341em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2659em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace mtight\" style=\"margin-right: 0.1952em\"><\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;f(n)&#061;\u0398(n)&#xff0c;\u547d\u4e2d\u7b2c2\u79cd\u60c5\u51b5&#xff08;k&#061;0&#xff09;&#xff0c;\u5f97 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        T<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        \u0398<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        log<\/p>\n<p>        \u2061<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       T(n)&#061;\\\\Theta(n\\\\log n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002\u4e0e\u8fed\u4ee3\u5c55\u5f00\u7684\u7ed3\u679c\u4e00\u81f4\u3002\u2705<\/p>\n<p>\u9884\u4e60\u9636\u6bb5\u5efa\u8bae\u4ee5\u8fed\u4ee3\u5c55\u5f00\u6cd5\u4e3a\u4e3b&#xff08;\u8003\u8bd5\u8981\u6c42\u5199\u8fc7\u7a0b&#xff09;&#xff0c;\u4e3b\u5b9a\u7406\u4f5c\u4e3a\u5feb\u901f\u9a8c\u7b97\u5de5\u5177\u3002<\/p>\n<h4>18.5 \u5e38\u89c1\u9012\u63a8\u5f0f\u901f\u67e5\u8868<\/h4>\n<table>\n<tr>\u9012\u63a8\u5173\u7cfb\u5f0f\u89e3\u5178\u578b\u7b97\u6cd5<\/tr>\n<tbody>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;T(n-1)&#043;O(1)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u7ebf\u6027\u9012\u5f52&#xff08;\u6c42n!\u3001\u8868\u63d2\u5165\u6392\u5e8f&#xff09;<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;T(n-1)&#043;O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>            n<\/p>\n<p>            2<\/p>\n<p>           )<\/p>\n<p>          O(n^2)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u6bcf\u5c42\u505a\u7ebf\u6027\u5de5\u4f5c\u7684\u9012\u51cf\u9012\u5f52<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           2<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>           2<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;2T(n\/2)&#043;O(1)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u4e8c\u53c9\u6811\u904d\u5386<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           2<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>           2<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;2T(n\/2)&#043;O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(n\\\\log n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u5f52\u5e76\u6392\u5e8f\u3001\u5feb\u901f\u6392\u5e8f&#xff08;\u5e73\u5747&#xff09;<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>           2<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;T(n\/2)&#043;O(1)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(\\\\log n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u4e8c\u5206\u67e5\u627e<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \/<\/p>\n<p>           2<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;T(n\/2)&#043;O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>          O(n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u6bcf\u5c42\u7ebf\u6027\u3001\u89c4\u6a21\u51cf\u534a&#xff08;\u5982\u9009\u62e9\u95ee\u9898\u7684\u4e00\u79cd\u89e3\u6cd5&#xff09;<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>           &#061;<\/p>\n<p>           2<\/p>\n<p>           T<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           &#043;<\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          T(n)&#061;2T(n-1)&#043;O(1)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>            2<\/p>\n<p>            n<\/p>\n<p>           )<\/p>\n<p>          O(2^n)<\/p>\n<p>       <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/td>\n<td>\u6c49\u8bfa\u5854<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u5341\u4e5d\u3001\u7a7a\u95f4\u590d\u6742\u5ea6\u5206\u6790<\/h3>\n<h4>19.1 \u57fa\u672c\u6982\u5ff5<\/h4>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u7684\u5b58\u50a8\u91cf\u5305\u62ec\u5f62\u53c2\u6240\u5360\u7a7a\u95f4\u548c\u4e34\u65f6\u53d8\u91cf\u6240\u5360\u7a7a\u95f4\u3002\u5728\u8fdb\u884c\u5b58\u50a8\u7a7a\u95f4\u5206\u6790\u65f6&#xff0c;\u53ea\u8003\u5bdf\u4e34\u65f6\u53d8\u91cf\u6240\u5360\u7a7a\u95f4\u3002<\/p>\n<p>\u4e3a\u4ec0\u4e48\u53ea\u7b97\u4e34\u65f6\u53d8\u91cf&#xff1f;\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u672c\u8eab\u5360\u7528\u7684\u7a7a\u95f4\u662f\u95ee\u9898\u56fa\u6709\u7684&#xff0c;\u4e0d\u56e0\u7b97\u6cd5\u800c\u5f02&#xff1b;\u7b97\u6cd5\u5206\u6790\u8981\u8861\u91cf\u7684\u662f&#034;\u7b97\u6cd5\u672c\u8eab\u989d\u5916\u9700\u8981\u591a\u5c11\u7a7a\u95f4&#034;\u3002<\/p>\n<p>\u7a7a\u95f4\u590d\u6742\u5ea6\u662f\u5bf9\u4e00\u4e2a\u7b97\u6cd5\u5728\u8fd0\u884c\u8fc7\u7a0b\u4e2d\u4e34\u65f6\u5360\u7528\u7684\u5b58\u50a8\u7a7a\u95f4\u5927\u5c0f\u7684\u91cf\u5ea6&#xff0c;\u4e00\u822c\u4e5f\u4f5c\u4e3a\u95ee\u9898\u89c4\u6a21 n \u7684\u51fd\u6570&#xff0c;\u4ee5\u6570\u91cf\u7ea7\u5f62\u5f0f\u7ed9\u51fa&#xff0c;\u8bb0\u4f5c&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         S<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>         \u3001<\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>         \u00a0\u6216\u00a0<\/p>\n<p>         \u0398<\/p>\n<p>         (<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         )<\/p>\n<p>        S(n)&#061;O(g(n))\u3001\\\\Omega(g(n)) \\\\text{ \u6216 } \\\\Theta(g(n))<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><span class=\"mord cjk_fallback\">\u3001<\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord cjk_fallback\">\u6216<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord\">\u0398<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">))<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5176\u4e2d\u6e10\u8fdb\u7b26\u53f7\u7684\u542b\u4e49\u4e0e\u65f6\u95f4\u590d\u6742\u5ea6\u4e2d\u7684\u542b\u4e49\u5b8c\u5168\u76f8\u540c\u3002<\/p>\n<h4>19.2 \u9759\u6001\u5206\u6790\u4e0e\u52a8\u6001\u5206\u6790<\/h4>\n<p>\u6839\u636e\u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b\u4e2d\u5bf9\u5b58\u50a8\u7a7a\u95f4\u7684\u4f7f\u7528\u65b9\u5f0f&#xff0c;\u53ef\u4ee5\u628a\u7a7a\u95f4\u590d\u6742\u6027\u5206\u6790\u5206\u6210\u4e24\u79cd&#xff1a;<\/p>\n<p>&#xff08;1&#xff09;\u9759\u6001\u5206\u6790<\/p>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u9759\u6001\u4f7f\u7528\u7684\u5b58\u50a8\u7a7a\u95f4&#xff0c;\u79f0\u4e3a\u9759\u6001\u7a7a\u95f4\u3002\u9759\u6001\u5206\u6790\u7684\u65b9\u6cd5\u6bd4\u8f83\u5bb9\u6613&#xff0c;\u53ea\u8981\u6c42\u51fa\u7b97\u6cd5\u4e2d\u4f7f\u7528\u7684\u6240\u6709\u53d8\u91cf\u7684\u7a7a\u95f4&#xff0c;\u518d\u6298\u5408\u6210\u591a\u5c11\u7a7a\u95f4\u5b58\u50a8\u5355\u4f4d\u5373\u53ef\u3002<\/p>\n<p>\u9759\u6001\u7a7a\u95f4\u5728\u7f16\u8bd1\/\u8fdb\u5165\u51fd\u6570\u65f6\u5c31\u4e00\u6b21\u6027\u5206\u914d\u597d&#xff0c;\u5927\u5c0f\u56fa\u5b9a\u4e0d\u53d8&#xff0c;\u5982\u666e\u901a\u5c40\u90e8\u53d8\u91cf int i, maxi;\u3002<\/p>\n<p>&#xff08;2&#xff09;\u52a8\u6001\u5206\u6790<\/p>\n<p>\u4e00\u4e2a\u7b97\u6cd5\u5728\u6267\u884c\u8fc7\u7a0b\u4e2d&#xff0c;\u5fc5\u987b\u4ee5\u52a8\u6001\u65b9\u5f0f\u5206\u914d\u7684\u5b58\u50a8\u7a7a\u95f4\u662f\u6307\u5728\u7b97\u6cd5\u6267\u884c\u8fc7\u7a0b\u4e2d\u5206\u914d\u7684\u7a7a\u95f4&#xff0c;\u79f0\u4e3a\u52a8\u6001\u7a7a\u95f4\u3002\u52a8\u6001\u7a7a\u95f4\u4e3b\u8981\u662f\u5b58\u50a8\u4e2d\u95f4\u7ed3\u679c\u6216\u64cd\u4f5c\u5355\u5143\u6240\u5360\u7528\u7a7a\u95f4\u3002<\/p>\n<p>\u5982 malloc\/new \u5206\u914d\u7684\u6570\u7ec4\u3001\u9012\u5f52\u8c03\u7528\u4ea7\u751f\u7684 \u5de5\u4f5c\u8bb0\u5f55&#xff08;\u6808\u5e27&#xff09; \u7b49&#xff0c;\u5176\u5927\u5c0f\u53ef\u80fd\u968f\u6267\u884c\u8fc7\u7a0b\u53d8\u5316\u3002<\/p>\n<h4>19.3 \u793a\u4f8b&#xff1a;\u6c42\u6700\u5927\u503c\u7684\u7b97\u6cd5<\/h4>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">max<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> maxi <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span>                    <span class=\"token comment\">\/\/ \u4e34\u65f6\u53d8\u91cf&#xff1a;i \u548c maxi<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> i <span class=\"token operator\">&lt;<\/span> n<span class=\"token punctuation\">;<\/span> i<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">)<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&gt;<\/span> a<span class=\"token punctuation\">[<\/span>maxi<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n            maxi <span class=\"token operator\">&#061;<\/span> i<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> a<span class=\"token punctuation\">[<\/span>maxi<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u51fd\u6570\u4f53\u5185\u5206\u914d\u7684\u53d8\u91cf\u7a7a\u95f4\u4e3a\u4e34\u65f6\u7a7a\u95f4&#xff0c;\u4e0d\u8ba1\u5f62\u53c2\u5360\u7528\u7684\u7a7a\u95f4\u3002\u8fd9\u91cc\u4ec5\u8ba1 i\u3001maxi \u4e24\u4e2a\u53d8\u91cf\u7684\u7a7a\u95f4&#xff08;\u5404\u4e00\u4e2a\u6574\u578b\u5355\u5143&#xff09;&#xff0c;\u4e0e\u95ee\u9898\u89c4\u6a21 n \u65e0\u5173&#xff0c;\u6240\u4ee5\u5176\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          O<\/p>\n<p>          (<\/p>\n<p>          1<\/p>\n<p>          )<\/p>\n<p>         O(1)<\/p>\n<p>      <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u26a0\ufe0f \u6570\u7ec4 a[] \u672c\u8eab\u662f\u8f93\u5165\u6570\u636e&#xff0c;\u5176 O(n) \u7684\u7a7a\u95f4\u4e0d\u7b97\u5728\u8be5\u7b97\u6cd5\u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u91cc\u3002<\/p>\n<h4>19.4 \u3010\u4f8b1-7\u3011\u975e\u9012\u5f52\u7b97\u6cd5\u7684\u7a7a\u95f4\u590d\u6742\u5ea6<\/h4>\n<p>\u5206\u6790\u4f8b1-5\u7b97\u6cd5\u7684\u7a7a\u95f4\u590d\u6742\u5ea6&#xff1a;<\/p>\n<p><span class=\"token keyword\">void<\/span> <span class=\"token function\">func<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> i <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> k <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">100<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;&#061;<\/span> n<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        k<span class=\"token operator\">&#043;&#043;<\/span><span class=\"token punctuation\">;<\/span><br \/>\n        i <span class=\"token operator\">&#043;&#061;<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u89e3&#xff1a;\u8be5\u7b97\u6cd5\u662f\u4e00\u4e2a\u975e\u9012\u5f52\u7b97\u6cd5&#xff0c;\u5176\u4e2d\u53ea\u4e34\u65f6\u5206\u914d\u4e86 i\u3001k \u4e24\u4e2a\u53d8\u91cf\u7684\u7a7a\u95f4&#xff0c;\u5b83\u4e0e\u95ee\u9898\u89c4\u6a21 n \u65e0\u5173&#xff0c;\u6240\u4ee5\u5176\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>       O(1)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5373\u8be5\u7b97\u6cd5\u4e3a\u539f\u5730\u5de5\u4f5c\u7b97\u6cd5&#xff08;in-place algorithm&#xff09;\u3002<\/p>\n<p>&#x1f4a1; \u539f\u5730\u5de5\u4f5c\u7b97\u6cd5&#xff1a;\u6307\u6267\u884c\u7b97\u6cd5\u6240\u9700\u7684\u8f85\u52a9\u5b58\u50a8\u7a7a\u95f4\u662f\u5e38\u91cf\u7ea7&#xff08;\u4e0e\u8f93\u5165\u89c4\u6a21\u65e0\u5173&#xff09;\u7684\u7b97\u6cd5\u3002\u51e1\u662f\u53ea\u7528\u4e86\u56fa\u5b9a\u51e0\u4e2a\u5c40\u90e8\u53d8\u91cf\u3001\u6ca1\u6709\u989d\u5916\u6570\u7ec4\/\u9012\u5f52\u6808\u7684\u7b97\u6cd5&#xff0c;\u7a7a\u95f4\u590d\u6742\u5ea6\u90fd\u662f O(1)\u3002<\/p>\n<h4>19.5 \u3010\u4f8b1-8\u3011\u9012\u5f52\u7b97\u6cd5\u7684\u7a7a\u95f4\u590d\u6742\u5ea6<\/h4>\n<p>\u9898\u76ee&#xff1a;\u6709\u5982\u4e0b\u9012\u5f52\u7b97\u6cd5&#xff0c;\u5206\u6790\u8c03\u7528 maxelem(a, 0, n-1) \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u3002<\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">maxelem<\/span><span class=\"token punctuation\">(<\/span><span class=\"token keyword\">int<\/span> a<span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> i<span class=\"token punctuation\">,<\/span> <span class=\"token keyword\">int<\/span> j<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token punctuation\">{<\/span><br \/>\n    <span class=\"token keyword\">int<\/span> mid <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&#043;<\/span> j<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token number\">2<\/span><span class=\"token punctuation\">,<\/span> max1<span class=\"token punctuation\">,<\/span> max2<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>i <span class=\"token operator\">&lt;<\/span> j<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token punctuation\">{<\/span><br \/>\n        max1 <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">maxelem<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> i<span class=\"token punctuation\">,<\/span> mid<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span>        <span class=\"token comment\">\/\/ \u9012\u5f52\u6c42\u5de6\u534a\u6700\u5927\u503c<\/span><br \/>\n        max2 <span class=\"token operator\">&#061;<\/span> <span class=\"token function\">maxelem<\/span><span class=\"token punctuation\">(<\/span>a<span class=\"token punctuation\">,<\/span> mid <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> j<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span>    <span class=\"token comment\">\/\/ \u9012\u5f52\u6c42\u53f3\u534a\u6700\u5927\u503c<\/span><br \/>\n        <span class=\"token keyword\">return<\/span> <span class=\"token punctuation\">(<\/span>max1 <span class=\"token operator\">&gt;<\/span> max2<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">?<\/span> max1 <span class=\"token operator\">:<\/span> max2<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><br \/>\n    <span class=\"token keyword\">else<\/span> <span class=\"token keyword\">return<\/span> a<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">;<\/span>                     <span class=\"token comment\">\/\/ \u9012\u5f52\u51fa\u53e3&#xff1a;\u53ea\u5269\u4e00\u4e2a\u5143\u7d20<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<p>\u89e3&#xff1a;\u6267\u884c\u8be5\u9012\u5f52\u7b97\u6cd5\u9700\u8981\u591a\u6b21\u8c03\u7528\u81ea\u8eab&#xff0c;\u6bcf\u6b21\u8c03\u7528\u53ea\u4e34\u65f6\u5206\u914d 3 \u4e2a\u6574\u578b\u53d8\u91cf&#xff08;mid\u3001max1\u3001max2&#xff09;\u7684\u7a7a\u95f4&#xff0c;\u5373\u6bcf\u5c42 O(1)\u3002<\/p>\n<p>\u5173\u952e\u70b9&#xff1a;\u9012\u5f52\u7b97\u6cd5\u7684\u7a7a\u95f4 &#061; \u9012\u5f52\u5de5\u4f5c\u6808\u4e2d\u540c\u65f6\u5b58\u5728\u7684\u6808\u5e27\u7a7a\u95f4\u4e4b\u548c\u3002\u867d\u7136\u5de6\u53f3\u4e24\u4e2a\u9012\u5f52\u8c03\u7528\u5148\u540e\u53d1\u751f&#xff0c;\u4f46\u7531\u4e8e max1 &#061; maxelem(&#8230;) \u7684\u7ed3\u679c\u8981\u4fdd\u7559\u5728\u6808\u5e27\u4e2d\u7b49\u5f85\u53f3\u534a\u8fb9\u7b97\u5b8c\u540e\u4f7f\u7528&#xff0c;\u9012\u5f52\u8c03\u7528\u94fe\u4e0a\u7684\u6808\u5e27\u662f\u53e0\u52a0\u7684\u3002<\/p>\n<p>\u8bbe\u8c03\u7528 maxelem(a, 0, n-1) \u7684\u7a7a\u95f4\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        S<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        )<\/p>\n<p>       S(n)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6709\u9012\u63a8\u5173\u7cfb&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         S<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>          {<\/p>\n<p>               O<\/p>\n<p>               (<\/p>\n<p>               1<\/p>\n<p>               )<\/p>\n<p>               n<\/p>\n<p>               &#061;<\/p>\n<p>               1<\/p>\n<p>               2<\/p>\n<p>               S<\/p>\n<p>               (<\/p>\n<p>               n<\/p>\n<p>               \/<\/p>\n<p>               2<\/p>\n<p>               )<\/p>\n<p>               &#043;<\/p>\n<p>               O<\/p>\n<p>               (<\/p>\n<p>               1<\/p>\n<p>               )<\/p>\n<p>               n<\/p>\n<p>               &gt;<\/p>\n<p>               1<\/p>\n<p>         S(n)&#061;\\\\begin{cases} O(1) &amp; n&#061;1 \\\\\\\\ 2S(n\/2)&#043;O(1) &amp; n&gt;1 \\\\end{cases} <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3em;vertical-align: -1.25em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">{<\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.69em\"><span class=\"\" style=\"top: -3.69em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -2.25em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.19em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"arraycolsep\" style=\"width: 1em\"><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.69em\"><span class=\"\" style=\"top: -3.69em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -2.25em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.19em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>&#xff08;\u8bf4\u660e&#xff1a;\u6559\u6750\u6b64\u5904\u6309 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         2<\/p>\n<p>         S<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         \/<\/p>\n<p>         2<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>        2S(n\/2)&#043;O(1)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5efa\u6a21\u5e76\u7ed9\u51fa O(n) \u7684\u7ed3\u8bba&#xff1b;\u76f4\u89c2\u4e0a\u4e5f\u53ef\u7406\u89e3\u4e3a\u9012\u5f52\u6811\u5171 n \u4e2a\u7ed3\u70b9\u3001\u6bcf\u4e2a\u7ed3\u70b9 O(1) \u7684\u6808\u5e27\u7a7a\u95f4&#xff0c;\u603b\u548c\u4e3a O(n)\u3002\u53e6\u4e00\u79cd\u66f4\u7cbe\u7ec6\u7684\u5206\u6790\u8ba4\u4e3a\u9012\u5f52\u6df1\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>        \\\\log_2 n<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>\u3001\u6bcf\u5c42 O(1)&#xff0c;\u6808\u7a7a\u95f4\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         log<\/p>\n<p>         \u2061<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        O(\\\\log n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>\u2014\u2014\u9884\u4e60\u9636\u6bb5\u4ee5\u6559\u6750\u7ed3\u8bba O(n) \u4e3a\u51c6&#xff0c;\u8003\u8bd5\u6309\u6559\u6750\u5efa\u6a21\u7b54\u9898\u3002&#xff09;<\/p>\n<p>\u8fed\u4ee3\u5c55\u5f00\u6c42\u89e3&#xff08;O(1) \u8bb0\u4e3a\u5e38\u65701&#xff09;&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>             S<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             )<\/p>\n<p>             &#061;<\/p>\n<p>             2<\/p>\n<p>             S<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>             2<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             1<\/p>\n<p>             &#061;<\/p>\n<p>             2<\/p>\n<p>              [<\/p>\n<p>              2<\/p>\n<p>              S<\/p>\n<p>              (<\/p>\n<p>              n<\/p>\n<p>              \/<\/p>\n<p>               2<\/p>\n<p>               2<\/p>\n<p>              )<\/p>\n<p>              &#043;<\/p>\n<p>              1<\/p>\n<p>              ]<\/p>\n<p>             &#043;<\/p>\n<p>             1<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>             S<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             1<\/p>\n<p>             &#043;<\/p>\n<p>              2<\/p>\n<p>              1<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              3<\/p>\n<p>             S<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>              2<\/p>\n<p>              3<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             1<\/p>\n<p>             &#043;<\/p>\n<p>              2<\/p>\n<p>              1<\/p>\n<p>             &#043;<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>             &#061;<\/p>\n<p>             \u22ef<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              k<\/p>\n<p>             S<\/p>\n<p>             (<\/p>\n<p>             n<\/p>\n<p>             \/<\/p>\n<p>              2<\/p>\n<p>              k<\/p>\n<p>             )<\/p>\n<p>             &#043;<\/p>\n<p>             1<\/p>\n<p>             &#043;<\/p>\n<p>              2<\/p>\n<p>              1<\/p>\n<p>             &#043;<\/p>\n<p>              2<\/p>\n<p>              2<\/p>\n<p>             &#043;<\/p>\n<p>             \u22ef<\/p>\n<p>             &#043;<\/p>\n<p>              2<\/p>\n<p>               k<\/p>\n<p>               \u2212<\/p>\n<p>               1<\/p>\n<p>             (<\/p>\n<p>             \u8bbe\u00a0<\/p>\n<p>             n<\/p>\n<p>             &#061;<\/p>\n<p>              2<\/p>\n<p>              k<\/p>\n<p>             ,<\/p>\n<p>             \u00a0\u5373\u00a0<\/p>\n<p>             k<\/p>\n<p>             &#061;<\/p>\n<p>               log<\/p>\n<p>               \u2061<\/p>\n<p>              2<\/p>\n<p>             n<\/p>\n<p>             )<\/p>\n<p>         \\\\begin{align*} S(n) &amp;&#061; 2S(n\/2)&#043;1 \\\\\\\\ &amp;&#061; 2\\\\left[2S(n\/2^2)&#043;1\\\\right]&#043;1 &#061; 2^2S(n\/2^2)&#043;1&#043;2^1 \\\\\\\\ &amp;&#061; 2^3S(n\/2^3)&#043;1&#043;2^1&#043;2^2 \\\\\\\\ &amp;&#061; \\\\cdots \\\\\\\\ &amp;&#061; 2^kS(n\/2^k)&#043;1&#043;2^1&#043;2^2&#043;\\\\cdots&#043;2^{k-1} \\\\qquad (\\\\text{\u8bbe } n&#061;2^k, \\\\text{ \u5373 } k&#061;\\\\log_2 n) \\\\end{align*} <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 7.6073em;vertical-align: -3.5537em\"><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-r\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 4.0537em\"><span class=\"\" style=\"top: -6.2137em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><span class=\"\" style=\"top: -4.6896em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><\/span><\/span><span class=\"\" style=\"top: -3.1654em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><\/span><\/span><span class=\"\" style=\"top: -1.6654em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><\/span><\/span><span class=\"\" style=\"top: -0.1063em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 3.5537em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 4.0537em\"><span class=\"\" style=\"top: -6.2137em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><span class=\"\" style=\"top: -4.6896em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size1\">[<\/span><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size1\">]<\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.1654em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -1.6654em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"minner\">\u22ef<\/span><\/span><\/span><span class=\"\" style=\"top: -0.1063em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord\"><span class=\"mord\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8641em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 2em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u8bbe<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord text\"><span class=\"mord\">\u00a0<\/span><span class=\"mord cjk_fallback\">\u5373<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 3.5537em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u4ee3\u5165 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        S<\/p>\n<p>        (<\/p>\n<p>        n<\/p>\n<p>        \/<\/p>\n<p>         2<\/p>\n<p>         k<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        S<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        O<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        )<\/p>\n<p>       S(n\/2^k)&#061;S(1)&#061;O(1)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5e76\u5229\u7528\u7b49\u6bd4\u6570\u5217\u6c42\u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u2211<\/p>\n<p>          t<\/p>\n<p>          &#061;<\/p>\n<p>          0<\/p>\n<p>          k<\/p>\n<p>          \u2212<\/p>\n<p>          1<\/p>\n<p>         2<\/p>\n<p>         t<\/p>\n<p>        &#061;<\/p>\n<p>         2<\/p>\n<p>         k<\/p>\n<p>        \u2212<\/p>\n<p>        1<\/p>\n<p>       \\\\sum_{t&#061;0}^{k-1}2^t &#061; 2^k &#8211; 1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.2887em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.989em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9324em;vertical-align: -0.0833em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         S<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         n<\/p>\n<p>         \u22c5<\/p>\n<p>         1<\/p>\n<p>         &#043;<\/p>\n<p>         (<\/p>\n<p>          2<\/p>\n<p>          k<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         n<\/p>\n<p>         &#043;<\/p>\n<p>         n<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>         &#061;<\/p>\n<p>         2<\/p>\n<p>         n<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         S(n) &#061; n\\\\cdot 1 &#043; (2^k &#8211; 1) &#061; n &#043; n &#8211; 1 &#061; 2n-1 &#061; O(n) <\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u22c5<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1491em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8991em\"><span class=\"\" style=\"top: -3.113em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u7ed3\u8bba&#xff1a;\u8c03\u7528 maxelem(a, 0, n-1) \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           O<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           )<\/p>\n<p>        \\\\boldsymbol{O(n)}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\"><span class=\"mord boldsymbol\" style=\"margin-right: 0.0319em\">O<\/span><span class=\"mopen mathbf\">(<\/span><span class=\"mord boldsymbol\">n<\/span><span class=\"mclose mathbf\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<h4>19.6 \u65f6\u95f4 vs \u7a7a\u95f4&#xff1a;\u9012\u5f52\u7684\u9690\u6027\u5f00\u9500<\/h4>\n<p>\u628a\u4f8b1-6&#xff08;\u5f52\u5e76\u6392\u5e8f&#xff09;\u548c\u4f8b1-8&#xff08;\u5206\u6cbb\u6c42\u6700\u5927\u503c&#xff09;\u653e\u5728\u4e00\u8d77\u770b&#xff0c;\u80fd\u4f53\u4f1a\u5230\u4e00\u4e2a\u91cd\u8981\u601d\u60f3&#xff1a;<\/p>\n<ul>\n<li>\u5206\u6cbb\u9012\u5f52\u7684\u65f6\u95f4&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         a<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         \/<\/p>\n<p>         b<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         f<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        T(n)&#061;aT(n\/b)&#043;f(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u7531&#034;\u5b50\u95ee\u9898\u5de5\u4f5c\u91cf &#043; \u5408\u5e76\u5de5\u4f5c\u91cf&#034;\u51b3\u5b9a&#xff1b;<\/li>\n<li>\u5206\u6cbb\u9012\u5f52\u7684\u7a7a\u95f4&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         S<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         a<\/p>\n<p>          T<\/p>\n<p>          \u2032<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         \/<\/p>\n<p>         b<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         g<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        S(n)&#061;aT&#039;(n\/b)&#043;g(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">S<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.0019em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7519em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u7531&#034;\u9012\u5f52\u5de5\u4f5c\u6808\u7684\u6808\u5e27\u7d2f\u79ef&#034;\u51b3\u5b9a&#xff1b;<\/li>\n<li>\u9012\u5f52\u603b\u662f\u6bd4\u7b49\u4ef7\u7684\u975e\u9012\u5f52\u5b9e\u73b0\u591a\u6d88\u8017\u6808\u7a7a\u95f4\u2014\u2014\u8fd9\u662f&#034;\u7528\u7a7a\u95f4\u6362\u4ee3\u7801\u7b80\u6d01\u6027&#034;\u7684\u5178\u578b\u4f8b\u5b50\u3002<\/li>\n<\/ul>\n<p>\u5bf9\u6bd4\u8bb0\u5fc6&#xff1a;<\/p>\n<table>\n<tr>\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u7a7a\u95f4\u590d\u6742\u5ea6<\/tr>\n<tbody>\n<tr>\n<td>\u987a\u5e8f\u67e5\u627e&#xff08;\u4f8b1-4&#xff09;<\/td>\n<td>\u6700\u597dO(1)\u3001\u6700\u574f\/\u5e73\u5747O(n)<\/td>\n<td>O(1)&#xff0c;\u539f\u5730<\/td>\n<\/tr>\n<tr>\n<td>\u4f8b1-5 while\u5faa\u73af<\/td>\n<td>O(n)<\/td>\n<td>O(1)&#xff0c;\u539f\u5730&#xff08;\u4f8b1-7&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u5f52\u5e76\u6392\u5e8f&#xff08;\u4f8b1-6&#xff09;<\/td>\n<td>O(n log n)<\/td>\n<td>\u6559\u6750\u6a21\u578bO(n)&#xff08;\u5408\u5e76\u9700\u8f85\u52a9\u6570\u7ec4&#043;\u9012\u5f52\u6808&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u5206\u6cbb\u6c42\u6700\u5927\u503c&#xff08;\u4f8b1-8&#xff09;<\/td>\n<td>O(n)&#xff08;\u53ef\u63a8&#xff1a;T(n)&#061;2T(n\/2)&#043;O(1)&#xff09;<\/td>\n<td>O(n)&#xff08;\u6559\u6750\u6a21\u578b&#xff09;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<h3>\u4e8c\u5341\u3001\u5168\u7ae0\u6613\u9519\u70b9\u5927\u76d8\u70b9<\/h3>\n<p>\u7ed3\u5408\u672c\u7ae0\u5168\u90e8\u4f8b\u9898&#xff0c;\u628a\u6700\u5bb9\u6613\u4e22\u5206\u7684\u5730\u65b9\u96c6\u4e2d\u5217\u4e00\u904d&#xff0c;\u8003\u524d\u626b\u4e00\u773c\u975e\u5e38\u6709\u7528&#xff1a;<\/p>\n<h4>20.1 \u6982\u5ff5\u7c7b\u6613\u9519\u70b9<\/h4>\n<li>\u8f93\u5165\u53ef\u4ee5\u662f0\u4e2a&#xff0c;\u8f93\u51fa\u5fc5\u987b\u81f3\u5c111\u4e2a\u3002\u9009\u62e9\u9898\u95ee&#034;\u7b97\u6cd5\u7684\u8f93\u5165\u4e2a\u6570&#034;&#xff0c;\u7b54\u6848\u662f&#034;\u96f6\u4e2a\u6216\u591a\u4e2a&#034;&#xff0c;\u4e0d\u662f&#034;\u81f3\u5c11\u4e00\u4e2a&#034;\u3002<\/li>\n<li>\u7b97\u6cd5\u5fc5\u987b\u6709\u7a77&#xff0c;\u7a0b\u5e8f\u4e0d\u4e00\u5b9a\u6709\u7a77\u3002\u64cd\u4f5c\u7cfb\u7edf\u3001\u670d\u52a1\u5668\u662f\u7a0b\u5e8f\u4f46\u4e0d\u662f\u7b97\u6cd5\u3002<\/li>\n<li>\u6709\u7a77\u6027 \u2260 \u53ef\u884c\u6027&#xff1a;\u6b7b\u5faa\u73af\u8fdd\u53cd\u6709\u7a77\u6027&#xff08;\u4f8b1-2\u63cf\u8ff01&#xff09;&#xff0c;\u9664\u96f6\u8fdd\u53cd\u53ef\u884c\u6027&#xff08;\u4f8b1-2\u63cf\u8ff02&#xff09;\u3002\u5224\u65ad\u53e3\u8bc0&#xff1a;\u6574\u4f53\u505c\u4e0d\u4e0b\u6765\u2192\u6709\u7a77\u6027&#xff1b;\u5355\u6b65\u505a\u4e0d\u4e86\u2192\u53ef\u884c\u6027\u3002<\/li>\n<li>\u5927O\u4e0a\u754c\u8d8a\u4f4e\u8d8a\u6709\u4ef7\u503c&#xff0c;\u5927\u03a9\u4e0b\u754c\u8d8a\u9ad8\u8d8a\u6709\u4ef7\u503c&#xff0c;\u65b9\u5411\u76f8\u53cd&#xff0c;\u522b\u8bb0\u6df7\u3002<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         )<\/p>\n<p>        O(n^2)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6bd4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          4<\/p>\n<p>         )<\/p>\n<p>        O(n^4)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">4<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6709\u4ef7\u503c&#xff1b;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>          n<\/p>\n<p>          2<\/p>\n<p>         )<\/p>\n<p>        \\\\Omega(n^2)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0641em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">n<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6bd4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a9<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>        \\\\Omega(n)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u03a9<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u6709\u4ef7\u503c\u3002<\/li>\n<li>f(n)&#061;\u0398(g(n)) \u5f53\u4e14\u4ec5\u5f53 f(n)&#061;O(g(n)) \u4e14 f(n)&#061;\u03a9(g(n))&#xff0c;\u0398 \u662f\u4e09\u8005\u4e2d\u6700\u7cbe\u786e\u7684\u3002<\/li>\n<li>\u7a7a\u95f4\u590d\u6742\u5ea6\u53ea\u7b97\u4e34\u65f6\u53d8\u91cf&#xff08;\u8f85\u52a9\u7a7a\u95f4&#xff09;&#xff0c;\u4e0d\u7b97\u8f93\u5165\u6570\u636e\u672c\u8eab\u548c\u5f62\u53c2\u3002max(int a[], int n) \u7684\u7a7a\u95f4\u590d\u6742\u5ea6\u662f O(1) \u800c\u4e0d\u662f O(n)\u3002<\/li>\n<li>\u5e73\u5747\u60c5\u51b5\u662f\u5e26\u6743\u5e73\u5747&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         A<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         \u2211<\/p>\n<p>         P<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         I<\/p>\n<p>         )<\/p>\n<p>        A(n)&#061;\\\\sum P(I)T(I)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6743\u91cd\u662f\u6982\u7387&#xff0c;\u4e0d\u662f\u7b80\u5355\u7b97\u672f\u5e73\u5747&#xff1b;\u4e14\u8981\u628a\u67e5\u627e\u5931\u8d25\u7684\u60c5\u5f62\u8ba1\u5165&#xff08;\u4f8b1-4\u7b2c(2)\u95ee&#xff09;\u3002<\/li>\n<h4>20.2 \u4ee3\u7801\u5206\u6790\u7c7b\u6613\u9519\u70b9<\/h4>\n<li>\u94fe\u8868\u5e8f\u53f7\u521d\u503c&#xff1a;\u5e26\u5934\u7ed3\u70b9\u94fe\u8868\u627e\u7b2c\u4e00\u4e2a\u6570\u636e\u7ed3\u70b9&#xff0c;\u5e8f\u53f7\u4ece1\u5f00\u59cb&#xff0c;i \u521d\u503c\u5e94\u4e3a1&#xff08;\u4f8b1-1&#xff09;\u3002<\/li>\n<li>\u5faa\u73af\u5224\u7a7a&#043;\u77ed\u8def\u987a\u5e8f&#xff1a;while (p !&#061; NULL &amp;&amp; p-&gt;data !&#061; x)\u2014\u2014\u5224\u7a7a\u5fc5\u987b\u5199\u5728\u524d\u9762&#xff0c;\u9760 &amp;&amp; \u77ed\u8def\u4fdd\u62a4\u3002<\/li>\n<li>\u5faa\u73af\u53d8\u91cf\u975e\u7ebf\u6027\u53d8\u5316&#xff1a;i &#043;&#061; 2 \u65f6\u7b2c m \u6b21\u540e i &#061; \u521d\u503c &#043; 2m&#xff1b;i *&#061; 2 \u65f6\u7b2c m \u6b21\u540e i &#061; \u521d\u503c \u00d7 2^m\u3002\u5148\u5199\u51fa\u901a\u9879\u518d\u89e3\u4e0d\u7b49\u5f0f&#xff08;\u4f8b1-5&#xff09;\u3002<\/li>\n<li>\u503c\u4f20\u9012\u5e26\u4e0d\u56de\u7ed3\u679c&#xff1a;\u8f93\u51fa\u578b\u53c2\u6570\u5fc5\u987b\u7528\u5f15\u7528&#xff08;int &amp;s&#xff09;\u6216\u6307\u9488&#xff0c;\u5426\u5219\u8c03\u7528\u65b9\u62ff\u4e0d\u5230\u8ba1\u7b97\u7ed3\u679c&#xff08;\u7b2c\u5341\u7ae0 fun \u51fd\u6570\u7684\u5751&#xff09;\u3002<\/li>\n<li>\u4e09\u91cd\u5faa\u73af\u6c42\u548c\u522b\u6570\u9519\u8fb9\u754c&#xff1a;j &lt;&#061; i \u4e0e j &lt; i\u3001k &lt; j \u4e0e k &lt;&#061; j \u5dee\u4e00\u6b21&#xff0c;\u5199\u6c42\u548c\u5f0f\u65f6\u4e0a\u4e0b\u9650\u8981\u9010\u4e00\u6838\u5bf9&#xff08;\u4f8b1-3&#xff09;\u3002<\/li>\n<h4>20.3 \u9012\u63a8\u6c42\u89e3\u7c7b\u6613\u9519\u70b9<\/h4>\n<li>\u9012\u63a8\u5f0f\u8981\u5199\u521d\u59cb\u6761\u4ef6&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         O<\/p>\n<p>         (<\/p>\n<p>         1<\/p>\n<p>         )<\/p>\n<p>        T(1)&#061;O(1)<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e0d\u80fd\u6f0f&#xff0c;\u5c55\u5f00\u89e6\u5e95\u65f6\u8981\u7528\u5b83\u3002<\/li>\n<li>\u5c55\u5f00\u627e\u89c4\u5f8b&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         2<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         \/<\/p>\n<p>         2<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         c<\/p>\n<p>         n<\/p>\n<p>        2T(n\/2)&#043;cn<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span> \u5c55\u5f00\u7b2c k \u6b65\u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          2<\/p>\n<p>          k<\/p>\n<p>         T<\/p>\n<p>         (<\/p>\n<p>         n<\/p>\n<p>         \/<\/p>\n<p>          2<\/p>\n<p>          k<\/p>\n<p>         )<\/p>\n<p>         &#043;<\/p>\n<p>         k<\/p>\n<p>         c<\/p>\n<p>         n<\/p>\n<p>        2^kT(n\/2^k)&#043;kcn<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mord mathnormal\">c<\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u7cfb\u6570\u3001\u89c4\u6a21\u3001\u7d2f\u8ba1\u5de5\u4f5c\u91cf\u4e09\u4e2a\u90e8\u5206\u540c\u6b65\u53d8\u5316&#xff0c;\u522b\u53ea\u5199\u4e00\u534a\u3002<\/li>\n<li>n&#061;2^k \u7684\u6362\u5143&#xff1a;\u5c55\u5f00\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         n<\/p>\n<p>         \/<\/p>\n<p>          2<\/p>\n<p>          k<\/p>\n<p>         &#061;<\/p>\n<p>         1<\/p>\n<p>        n\/2^k&#061;1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0991em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">n<\/span><span class=\"mord\">\/<\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u65f6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         k<\/p>\n<p>         &#061;<\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>        k&#061;\\\\log_2 n<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u6700\u7ec8\u7ed3\u679c\u91cc\u7684 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>           log<\/p>\n<p>           \u2061<\/p>\n<p>          2<\/p>\n<p>         n<\/p>\n<p>        \\\\log_2 n<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9386em;vertical-align: -0.2441em\"><\/span><span class=\"mop\"><span class=\"mop\">lo<span style=\"margin-right: 0.0139em\">g<\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.207em\"><span class=\"\" style=\"top: -2.4559em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2441em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">n<\/span><\/span><\/span><\/span><\/span> \u5c31\u662f\u8fd9\u4e48\u6765\u7684\u3002<\/li>\n<li>\u7b49\u6bd4\u6c42\u548c\u516c\u5f0f&#xff1a;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         1<\/p>\n<p>         &#043;<\/p>\n<p>         2<\/p>\n<p>         &#043;<\/p>\n<p>          2<\/p>\n<p>          2<\/p>\n<p>         &#043;<\/p>\n<p>         \u22ef<\/p>\n<p>         &#043;<\/p>\n<p>          2<\/p>\n<p>           k<\/p>\n<p>           \u2212<\/p>\n<p>           1<\/p>\n<p>         &#061;<\/p>\n<p>          2<\/p>\n<p>          k<\/p>\n<p>         \u2212<\/p>\n<p>         1<\/p>\n<p>        1&#043;2&#043;2^2&#043;\\\\cdots&#043;2^{k-1}&#061;2^k-1<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">2<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8974em;vertical-align: -0.0833em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8491em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9324em;vertical-align: -0.0833em\"><\/span><span class=\"mord\"><span class=\"mord\">2<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8491em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right: 0.0315em\">k<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span>&#xff08;\u4f8b1-8\u6700\u540e\u4e00\u6b65&#xff09;&#xff0c;\u7b49\u5dee\u6c42\u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>         i<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          2<\/p>\n<p>        \\\\sum_{i&#061;1}^{n}i&#061;\\\\frac{n(n&#043;1)}{2}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.104em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\">i<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.355em;vertical-align: -0.345em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.01em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.485em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u5e73\u65b9\u548c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          \u2211<\/p>\n<p>           i<\/p>\n<p>           &#061;<\/p>\n<p>           1<\/p>\n<p>          n<\/p>\n<p>          i<\/p>\n<p>          2<\/p>\n<p>         &#061;<\/p>\n<p>           n<\/p>\n<p>           (<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>           (<\/p>\n<p>           2<\/p>\n<p>           n<\/p>\n<p>           &#043;<\/p>\n<p>           1<\/p>\n<p>           )<\/p>\n<p>          6<\/p>\n<p>        \\\\sum_{i&#061;1}^n i^2 &#061; \\\\frac{n(n&#043;1)(2n&#043;1)}{6}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1138em;vertical-align: -0.2997em\"><\/span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position: relative;top: 0em\">\u2211<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8043em\"><span class=\"\" style=\"top: -2.4003em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i<\/span><span class=\"mrel mtight\">&#061;<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.2029em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2997em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">i<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.355em;vertical-align: -0.345em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.01em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">6<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.485em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight\">)<\/span><span class=\"mopen mtight\">(<\/span><span class=\"mord mtight\">2<\/span><span class=\"mord mathnormal mtight\">n<\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mtight\">1<\/span><span class=\"mclose mtight\">)<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span><\/span> \u5fc5\u987b\u719f\u8bb0\u3002<\/li>\n<hr \/>\n<h3>\u4e8c\u5341\u4e00\u3001\u8003\u70b9\u603b\u7ed3\u4e0e\u5b66\u4e60\u5efa\u8bae<\/h3>\n<h4>21.1 \u672c\u7ae0\u77e5\u8bc6\u6846\u67b6\u56fe&#xff08;\u6587\u5b57\u7248&#xff09;<\/h4>\n<p>\u7b97\u6cd5\u8bbe\u8ba1\u4e0e\u5206\u6790\u00b7\u7b2c\u4e00\u7ae0<br \/>\n\u2502<br \/>\n\u251c\u2500\u2500 1.1 \u6982\u8ff0<br \/>\n\u2502   \u251c\u2500\u2500 \u7b97\u6cd5\u5b9a\u4e49&#xff1a;\u8f93\u5165 \u2192 \u4e00\u7cfb\u5217\u6b65\u9aa4 \u2192 \u8f93\u51fa<br \/>\n\u2502   \u251c\u2500\u2500 \u7b97\u6cd5\u6b63\u786e\u6027&#xff1a;\u5bf9\u6bcf\u4e2a\u8f93\u5165\u5b9e\u4f8b\u8f93\u51fa\u6b63\u786e\u7ed3\u679c\u5e76\u505c\u6b62<br \/>\n\u2502   \u251c\u2500\u2500 \u8bbe\u8ba1\u516d\u5927\u76ee\u6807&#xff1a;\u6b63\u786e\u6027\/\u53ef\u4f7f\u7528\u6027\/\u53ef\u8bfb\u6027\/\u5065\u58ee\u6027\/\u9ad8\u6548\u7387\/\u4f4e\u5b58\u50a8\u91cf<br \/>\n\u2502   \u251c\u2500\u2500 \u4e94\u5927\u7279\u5f81&#xff08;Knuth&#xff09;&#xff1a;\u6709\u7a77\u6027\u3001\u786e\u5b9a\u6027\u3001\u8f93\u5165(0..n)\u3001\u8f93\u51fa(1..n)\u3001\u53ef\u884c\u6027<br \/>\n\u2502   \u251c\u2500\u2500 \u63cf\u8ff0\u65b9\u6cd5&#xff1a;\u81ea\u7136\u8bed\u8a00\u3001\u6d41\u7a0b\u56fe\u3001\u4f2a\u4ee3\u7801<br \/>\n\u2502   \u251c\u2500\u2500 \u4f8b1-1 \u94fe\u8868\u67e5\u627e\u627e\u832c&#xff08;\u521d\u503ci&#061;1\u3001\u5224\u7a7a\u77ed\u8def\u3001\u533a\u5206\u6210\u529f\u5931\u8d25&#xff09;<br \/>\n\u2502   \u251c\u2500\u2500 \u4f8b1-2 \u6b7b\u5faa\u73af\u2192\u6709\u7a77\u6027&#xff1b;\u9664\u96f6\u2192\u53ef\u884c\u6027<br \/>\n\u2502   \u251c\u2500\u2500 \u6848\u4f8b&#xff1a;6x&#043;5y&#043;4z&#061;50\u7a77\u4e3e&#xff1b;\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u7279\u5f81\u8fa8\u6790<br \/>\n\u2502   \u2514\u2500\u2500 \u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784&#xff1a;\u8054\u7cfb&#xff08;\u7ed3\u6784\u662f\u7b97\u6cd5\u57fa\u7840&#xff09;\/\u533a\u522b&#xff08;\u7ec4\u7ec7 vs \u6c42\u89e3&#xff09;<br \/>\n\u2502       \u6c42\u89e3\u6d41\u7a0b&#xff1a;\u5206\u6790\u95ee\u9898\u2192\u9009\u7ed3\u6784\u4e0e\u7b56\u7565\u2192\u63cf\u8ff0\u2192\u8bc1\u660e\u6b63\u786e\u6027\u2192\u5206\u6790<br \/>\n\u2502<br \/>\n\u2514\u2500\u2500 1.2 \u7b97\u6cd5\u7684\u5206\u6790<br \/>\n    \u251c\u2500\u2500 \u6784\u6210&#xff1a;\u63a7\u5236\u7ed3\u6784&#xff08;\u987a\u5e8f\/\u5206\u652f\/\u5faa\u73af&#xff09;&#043; \u539f\u64cd\u4f5c<br \/>\n    \u251c\u2500\u2500 \u65f6\u95f4\u590d\u6742\u5ea6<br \/>\n    \u2502   \u251c\u2500\u2500 \u6b65\u9aa4&#xff1a;\u627e\u57fa\u672c\u8bed\u53e5 \u2192 \u6c42f(n) \u2192 \u7528O\/\u03a9\/\u0398\u8868\u9636<br \/>\n    \u2502   \u251c\u2500\u2500 \u6e10\u8fdb\u7b26\u53f7&#xff1a;O(\u4e0a\u754c,\u8d8a\u4f4e\u8d8a\u6709\u4ef7\u503c)\u3001\u03a9(\u4e0b\u754c,\u8d8a\u9ad8\u8d8a\u6709\u4ef7\u503c)\u3001\u0398(\u540c\u9636,\u6700\u7cbe\u786e)<br \/>\n    \u2502   \u251c\u2500\u2500 \u591a\u9879\u5f0f&#xff1a;a\u2098n\u1d50&#043;\u2026&#043;a\u2080 \u2192 O(n\u1d50)\/\u03a9(n\u1d50)\/\u0398(n\u1d50)<br \/>\n    \u2502   \u251c\u2500\u2500 \u4f8b1-3 \u4e09\u91cd\u5faa\u73af \u2192 O(n\u00b3)<br \/>\n    \u2502   \u251c\u2500\u2500 \u4e09\u79cd\u60c5\u51b5&#xff1a;G(n)&#061;MIN\u3001W(n)&#061;MAX\u3001A(n)&#061;\u03a3P(I)T(I)<br \/>\n    \u2502   \u251c\u2500\u2500 \u4f8b1-4 \u987a\u5e8f\u67e5\u627e&#xff1a;G&#061;O(1)\u3001W&#061;O(n)\u3001A&#061;(n&#043;1)\/2&#xff1b;<br \/>\n    \u2502   \u2502        \u542b\u5931\u8d25\u6982\u7387q&#xff1a;A&#061;q(n&#043;1)\/2&#043;(1\u2212q)n&#xff0c;q&#061;1\/2\u65f6\u22483n\/4<br \/>\n    \u2502   \u251c\u2500\u2500 \u975e\u9012\u5f52&#xff1a;\u4f8b1-5 i&#043;&#061;2 \u2192 O(n)&#xff1b;\u53d8\u5f0f i*&#061;2 \u2192 O(log n)<br \/>\n    \u2502   \u2514\u2500\u2500 \u9012\u5f52&#xff1a;\u5efa\u9012\u63a8\u5f0f\u2192\u8fed\u4ee3\u5c55\u5f00<br \/>\n    \u2502            \u4f8b1-6 \u5f52\u5e76\u6392\u5e8f T(n)&#061;2T(n\/2)&#043;O(n) \u2192 O(nlog\u2082n)<br \/>\n    \u2514\u2500\u2500 \u7a7a\u95f4\u590d\u6742\u5ea6<br \/>\n        \u251c\u2500\u2500 \u53ea\u8ba1\u4e34\u65f6\u53d8\u91cf&#xff1b;\u9759\u6001\u7a7a\u95f4 vs \u52a8\u6001\u7a7a\u95f4<br \/>\n        \u251c\u2500\u2500 \u4f8b1-7 \u975e\u9012\u5f52\u4e24\u53d8\u91cf \u2192 O(1)&#xff08;\u539f\u5730\u5de5\u4f5c\u7b97\u6cd5&#xff09;<br \/>\n        \u2514\u2500\u2500 \u4f8b1-8 \u9012\u5f52maxelem S(n)&#061;2S(n\/2)&#043;O(1) \u2192 2n\u22121 \u2192 O(n)<\/p>\n<h4>21.2 \u9ad8\u9891\u9898\u578b\u9884\u6d4b<\/h4>\n<p>\u6839\u636e\u672c\u7ae0\u5185\u5bb9&#xff0c;\u8003\u8bd5\u5927\u6982\u7387\u51fa\u73b0\u4ee5\u4e0b\u9898\u578b&#xff1a;<\/p>\n<table>\n<tr>\u9898\u578b\u5bf9\u5e94\u4f8b\u9898\u5206\u503c\u6743\u91cd<\/tr>\n<tbody>\n<tr>\n<td>\u7b97\u6cd5\u7279\u5f81\u8fa8\u6790&#xff08;\u9009\u62e9\/\u586b\u7a7a&#xff09;<\/td>\n<td>\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u586b\u7a7a\u3001\u4f8b1-2<\/td>\n<td>\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<tr>\n<td>\u627e\u832c\u6539\u9519\u9898&#xff08;\u6b63\u786e\u6027&#043;\u5065\u58ee\u6027&#xff09;<\/td>\n<td>\u4f8b1-1<\/td>\n<td>\u2b50\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<tr>\n<td>\u7ed9\u4ee3\u7801\u6c42\u65f6\u95f4\u590d\u6742\u5ea6<\/td>\n<td>\u4f8b1-3\u3001\u4f8b1-5\u53ca\u5404\u79cd\u5faa\u73af\u53d8\u5f0f<\/td>\n<td>\u2b50\u2b50\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<tr>\n<td>\u4e09\u79cd\u60c5\u51b5\u5206\u6790&#043;\u5e73\u5747\u590d\u6742\u5ea6\u63a8\u5bfc<\/td>\n<td>\u4f8b1-4<\/td>\n<td>\u2b50\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<tr>\n<td>\u9012\u5f52\u7b97\u6cd5\u9012\u63a8\u5f0f\u5efa\u7acb\u4e0e\u6c42\u89e3<\/td>\n<td>\u4f8b1-6\u3001\u4f8b1-8<\/td>\n<td>\u2b50\u2b50\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<tr>\n<td>\u7a7a\u95f4\u590d\u6742\u5ea6\u5206\u6790&#xff08;\u542b\u9012\u5f52\u6808&#xff09;<\/td>\n<td>\u4f8b1-7\u3001\u4f8b1-8<\/td>\n<td>\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<tr>\n<td>\u7b80\u7b54&#xff1a;\u7b97\u6cd5\u4e0e\u6570\u636e\u7ed3\u6784\u7684\u5173\u7cfb\u3001\u6e10\u8fdb\u7b26\u53f7\u5b9a\u4e49<\/td>\n<td>\u7b2c2.4\u8282\u3001\u7b2c11\u8282\u3001\u7b2c13\u8282<\/td>\n<td>\u2b50\u2b50\u2b50<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>21.3 \u7ed9\u540c\u5b66\u4eec\u7684\u5b66\u4e60\u5efa\u8bae<\/h4>\n<li>\u52a8\u624b\u63a8\u5bfc&#xff0c;\u62d2\u7edd\u773c\u9ad8\u624b\u4f4e\u3002\u6e10\u8fdb\u5206\u6790\u7684\u6240\u6709\u9898\u76ee&#xff0c;\u65b9\u6cd5\u90fd\u53ea\u6709\u51e0\u62db&#xff08;\u627e\u57fa\u672c\u8bed\u53e5\u3001\u5199\u6c42\u548c\u5f0f\u3001\u5efa\u9012\u63a8\u5f0f\u3001\u8fed\u4ee3\u5c55\u5f00&#xff09;&#xff0c;\u4f46\u6bcf\u4e00\u62db\u90fd\u9700\u8981\u4eb2\u624b\u7b97\u51e0\u904d\u624d\u80fd\u5185\u5316\u3002\u5efa\u8bae\u628a\u4f8b1-3\u7684\u4e09\u91cd\u6c42\u548c\u3001\u4f8b1-6\u7684\u5f52\u5e76\u5c55\u5f00\u5404\u72ec\u7acb\u63a83\u904d\u3002<\/li>\n<li>\u80cc\u719f\u4e09\u5927\u6c42\u548c\u516c\u5f0f\u548c\u5e38\u89c1\u9012\u63a8\u5f0f\u901f\u67e5\u8868&#xff08;\u89c118.5\u8282&#xff09;&#xff0c;\u5b83\u4eec\u662f\u8ba1\u7b97\u9898\u7684&#034;\u4e58\u6cd5\u53e3\u8bc0&#034;\u3002<\/li>\n<li>\u5efa\u7acb&#034;\u590d\u6742\u5ea6\u76f4\u89c9&#034;\u3002\u770b\u5230\u5355\u5c42\u5faa\u73af\u60f3 O(n)&#xff0c;\u5d4c\u5957\u4e24\u5c42\u60f3 O(n\u00b2)&#xff0c;\u5faa\u73af\u53d8\u91cf\u500d\u589e\u60f3 O(log n)&#xff0c;&#034;\u4e00\u5206\u4e3a\u4e8c&#043;\u7ebf\u6027\u5408\u5e76&#034;\u60f3 O(n log n)&#xff0c;&#034;\u89c4\u6a21\u51cf\u4e00&#034;\u7684\u9012\u5f52\u60f3 O(n)\u2026\u2026\u505a\u9898\u524d\u5148\u731c\u7b54\u6848\u518d\u9a8c\u8bc1&#xff0c;\u8fdb\u6b65\u6700\u5feb\u3002<\/li>\n<li>\u91cd\u89c6\u6982\u5ff5\u8fa8\u6790\u9898\u3002\u6709\u7a77\u6027\/\u53ef\u884c\u6027\u3001O\/\u03a9\/\u0398\u3001\u9759\u6001\/\u52a8\u6001\u7a7a\u95f4\u3001\u6700\u597d\/\u6700\u574f\/\u5e73\u5747\u2014\u2014\u8fd9\u4e9b\u6982\u5ff5\u4e24\u4e24\u4e4b\u95f4\u7684\u533a\u522b\u90fd\u662f\u9009\u62e9\u9898\u548c\u586b\u7a7a\u9898\u7684\u5bcc\u77ff&#xff0c;\u5efa\u8bae\u81ea\u5df1\u5236\u4f5c\u5bf9\u6bd4\u5361\u7247\u3002<\/li>\n<li>\u5199\u4ee3\u7801\u65f6\u517b\u6210\u5206\u6790\u4e60\u60ef\u3002\u4eca\u540e\u6bcf\u5199\u4e00\u4e2a\u51fd\u6570&#xff0c;\u987a\u624b\u5728\u6ce8\u91ca\u91cc\u6807\u660e\u65f6\u95f4\u548c\u7a7a\u95f4\u590d\u6742\u5ea6\u3002\u8fd9\u4e0d\u4ec5\u662f\u672c\u8bfe\u7a0b\u7684\u8981\u6c42&#xff0c;\u66f4\u662f\u7ec8\u8eab\u53d7\u7528\u7684\u5de5\u7a0b\u7d20\u517b\u3002<\/li>\n<li>\u4e3a\u540e\u7eed\u7ae0\u8282\u94fa\u57ab\u3002\u7b2c\u4e00\u7ae0\u7684\u9012\u63a8\u5f0f\u6c42\u89e3\u65b9\u6cd5\u5c06\u76f4\u63a5\u7528\u4e8e\u7b2c\u4e8c\u7ae0\u9012\u5f52\/\u5206\u6cbb\u7b97\u6cd5\u7684\u5206\u6790&#xff1b;&#034;\u6700\u597d\/\u6700\u574f\/\u5e73\u5747&#034;\u6846\u67b6\u5c06\u7528\u4e8e\u6392\u5e8f\u548c\u67e5\u627e\u7b97\u6cd5\u7684\u8bc4\u4ef7\u3002\u57fa\u7840\u4e0d\u7262&#xff0c;\u5730\u52a8\u5c71\u6447\u3002<\/li>\n<h4>21.4 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